NO, most certainly not. The only “evidence” indicating that this was the case is a heinous letter from Mary Queen of Scots relating hearsay, backstairs gossip and downright lies. (See 2.2 Was Queen Elizabeth I a virgin?)
This faint murmur of a rumour, that was already ten years old when Mary tried to bring it to life was as follows: “... mesme le Comte d’Oxfort n’osoit ce rappointer auveques sa famme de peur de perdre la faveur qu’il esperoit recepvoir par vous fayre l’amour”- that means: “even the Earl of Oxford dared not reconcile himself with his wife for fear of losing the favour which he hoped to receive by courting you”. The expression: “par vous fayre l’amour” is not the same as “faire l’amour avec vous”. The first meaning to pay close attention to a person and to show affection, the second meaning to make love to them. (We are reminded of Hortensio's speech in “The Taming of the Shrew” (I/2); the love sick Hortensio says: “That so I may, by this device, at least / Have leave and leisure to make love to her / And unsuspected court her by herself.”) Mary Stuart insinuates that Oxford spoons incessantly over Elizabeth and is so scared of arousing Elizabeth's jelousy that he doesn't go home to his wife. Mary's source of information is a malicious gossip by the name of “Bess of Hardwick”, the wife of the Earl of Shrewsbury, Mary's jailer at Tutbury Castle.
For her part, the malevolent “Bess of Hardwick” (Elizabeth Talbot, Countess of Shrewsbury, received her information from her son, Gilbert Talbot, who, on 13th May 1573, wrote to his father:
“ My Lord of Oxford is lately grown into great credit, for the Queen's Majesty delighteth more in his personage and his dancing and valiantness than any other. I think Sussex doth back him all that he can; if it were not for his fickle head, he would pass any of them shortly. My Lady Burghley unwisely has declared herself, as it were, jealous, which is come to the Queen's ear, whereat she has been not a little offended with her, but now she is reconciled again.”
At the period of time in question Lord Burghley (Oxford's father in law) and his wife were both at court. There was a certain tension between Oxford and his mother in law. She was jealous of her son in law, perhaps because she and her daughter looked a little colourless in the shadow of his wit and his rhetorical skills.
Lord Burghley was not only the Lord High Treasurer, he also regarded himself as being the guardian of morals and propriety in England. If he'd have thought for a second that Oxford was cheating on his daughter with the Queen, he'd have had his guts for garters.
Mildred Cooke, Lady Burghley (1526-1589)
Only once do we find Oxford's name in association with court intrigues: In a letter from 9 October 1572 to his friend, Christopher Hatton, advising him how to win his way into the affections of Elizabeth, the courtier Edward Dyer writes: “Marry thus much I would advise you to remember, that you use no words of disgrace or reproach towards him [Chm =Lord Great Chamberlain Oxford] to any; that he, being the less provoked, may sleep, thinking all safe, while you do awake and attend your advantages.”- Other letters from Christopher Hatton reveal that he and the slightly younger Earl of Oxford were rivals. Hatton wrote a letter to Queen Elizabeth in June 1573: “God bless you forever; the branch of the sweetest bush I will wear and bear to my life’s end: God witness I feign it not. It is a gracious favor most dear and welcome unto me: reserve it to the Sheep [Hatton], he hath no tooth to bite, where the Boar's [Oxford] tusk may both raze and tear.”
On 12. November 1577 Christopher Hatton is knighted by the Queen. Sir John Perrott (1530-1592) teases the young man saying that he found his way into royal favour by means of the galliard. Later, William Shakespeare makes a humorous reference to this taunt in “Twelfth Night” I/3:
SIR TOBY. What is thy excellence in a galliard, knight?
AGUECHEEK. Faith, I can cut a caper.
SIR TOBY. And I can cut the mutton to't. [..] Is it a world to hide virtues in? I did think, by the excellent constitution of thy leg, it was form'd under the star of a galliard.
AGUECHEEK. Ay, 'tis strong, and it does indifferent well in flame-colour'd stock.
Alas we digress. We've slipped to further proof that Oxford was Shake-speare rather than stick to the point at hand. It's tiresome to stick to the question of Oxford's being Elizabeth's lover, simply because the accusation is so ludicrous.
Sir Edward Dyer to Sir Christopher Hatton, 9 October 1572:
Sir, After my departure from you, thinking upon your case as my dear friend, I thought good to lay before you mine opinion in writing somewhat more at large than at my last conference I did speak....
The best and soundest way in mine opinion is, to put on another mind; to use your suits towards her Majesty in words, behaviour, and deeds; to acknowledge your duty, declaring the reverence which in heart you bear, and never seem deeply to condemn her frailties, but rather joyfully to commend such things as should be in her, as though they were in her indeed; hating my Lord Chm [Chamberlain =Oxford?] in the Queen's understanding for affection's sake, and blaming him openly for seeking the Queen's favour. For though in the beginning when her Majesty sought you (after her good manlier), she did bear with rugged dealing of yours, until she had what she fancied, yet now, after satiety and fulness, it [i.e. rugged dealing] will rather hurt than help you; whereas, behaving yourself as I said before, your place shall keep you in worship, your presence in favour, your followers will stand to you, at least you shall have no bold enemies, and you shall dwell in the ways to take all advantage wisely, and honestly to serve your turn at times. Marry thus much I would advise you to remember, that you use no words of disgrace or reproach towards him to any; that he, being the less provoked, may sleep, thinking all safe, while you do awake and attend your advantages.
Otherwise you shall, as it were, warden him and keep him in order; and he will make the Queen think that he beareth all for her sake, which will be as a merit in her sight; and the pursuing of his revenge shall be just in all men's opinions, by what means soever he and his friends shall ever be able ...
BL, Harleian MS., 787, fol. 88; quoted by Frederick Chamberlin, The Private Character of Queen Elizabeth, London 1921, pp. 181-2.
Sir Christopher Hatton (1540-1591)
Letter of Gilbert Talbot to his father, George Talbot, Earl of Shrewsbury, 11 May 1573:
My most humble duty remembered, right honourable my singular good Lord and father, because of the convenience of the bearer hereof I have thought good to advertise your Lordship of the estate of some here at the Court as near as I have learned by my daily experience.
My Lord Treasurer, even after the old manner, dealeth with matters of the state only, and beareth himself very uprightly. My Lord Leicester is very much with her Majesty, and she shows the same great good affection to him that she was wont; of late he has endeavoured to please her more than heretofore. There are two sisters now in the court that are very far in love with him, as they have been long, my Lady Sheffield and Frances Howard. They (of like striving who shall love him better) are at great wars together, and the Queen thinketh not well of them, and not the better of him; by this means there are spies over him. My Lord of Sussex goes with the tide, and helps to back others, but his own credit is sober, considering his estate; he is very diligent in his office, and takes great pains. My Lord of Oxford is lately grown into great credit, for the Queen's Majesty delighteth more in his personage and his dancing and valiantness than any other. I think Sussex doth back him all that he can; if it were not for his fickle head, he would pass any of them shortly. My Lady Burghley unwisely has declared herself, as it were, jealous, which is come to the Queen's ear, whereat she has been not a little offended with her, but now she is reconciled again. At all these love matters my Lord Treasurer winketh, and will not meddle any way. Hatton is sick still; it is thought he will very hardly recover his disease, for it is doubted it is in his kidneys. The Queen goeth almost every day to see how he doth. Now are there devices (chiefly by Leicester, as I suppose, and not without Burghley's knowledge) how to make Mr Edward Dyer as great as ever was Hatton, for now in this time of Hatton's sickness the time is convenient. It is brought thus to pass. Dyer lately was sick of a consumption, in great danger, and as your Lordship knows he has been in displeasure these two years. It was made the Queen believe that his sickness came because of the continuance of her displeasure towards him, so that unless she would forgive him, he was like not to recover, and hereupon her Majesty has forgiven him and sent unto him a very comfortable message; now he is recovered again, and this is the beginning of this device. These things I learn of such young fellows as myself. [...]
Roulsdon [Roulstone] hath written to your Lordship, as he saith, by this bearer, he trusteth to your Lordship's satisfaction. I have been very importunate of him for the present payment of his debt to your Lordship. He cannot any ways make shift for money unless he sell land, which he vows to do rather than to purchase your Lordship's displeasure.
Lodge, Edmund, Illustrations of British History, Biography, and Manners in the Reigns of Henry VIII, Edward VI, Mary, Elizabeth, & James I, vol. II (London: 1838) pp. 16-21.
A Letter from the Queen of Scots to Queen Elizabeth, 1584:
According to what I promised you and you have since desired, I declare to you now, with regret that such things should be brought into question but very sincerely and without any anger, which I call my God to witness, that the Countess of Shrewsbury said to me about you what follows as nearly as possible in these terms, to the greater part of which I protest I answered, rebutting the said lady for believing or speaking so licentiously of you as a thing which I did not at all believe and do not now believe, knowing the disposition of the Countess and by what spirit she was then urged on against you.
Firstly that one [Robert Dudley, Earl of Leicester] to whom she said you had made a promise of marriage before a lady of your chamber had lain numberless times with you ... and that you would never wish to lose the liberty of making love and gratifying yourself with new lovers, regretting this, said she, that you would not content yourself with Master Hatton and another of this kingdom... That even the Earl of Oxford dared not reconcile himself with his wife for fear of losing the favour which he hoped to receive by courting you [que mesme le Comte d’Oxfort n’osoit ce rappointer auveques sa famme de peur de perdre la faveur qu’il esperoit recepvoir par vous fayre l’amour].
Quoted by Chamberlin, Frederick: The Private Character of Queen Elizabeth, London 1921, p. 167