I have seen the movie "Mrs. Brown" and I think there is probably some truth to what was portrayed. The movie did not show any affair between the two, and is actually not what the movie is about.
John Brown was a Scot in service at Balmoral Castle when the Queen and Prince Albert purchased it in 1853. He was highly regarded by the Queen, especially after Albert's death.
I believe he is the one who pulled her out of her depression over her husband's death, and by all accounts he was highly protective of her and was a very plain-spoken man, which almost all the court resented.. except of course, for the Queen.
I think what they shared was a deep and abiding platonic love and friendship.. one the Queen desperately needed in the years of her widowhood. She was certainly consumed with grief when he died in 1883, and in a letter to Viscount Cranbrook, she writes:
"Perhaps never in history was there so strong and true an attachment, so warm and loving a friendship between the sovereign and servant ... Strength of character as well as power of frame – the most fearless uprightness, kindness, sense of justice, honesty, independence and unselfishness combined with a tender, warm heart ... made him one of the most remarkable men. The Queen feels that life for the second time is become most trying and sad to bear deprived of all she so needs ... the blow has fallen too heavily not to be very heavily felt..."
That says it all.
But her attachment to Mr. Brown was such that when she died, Queen Victoria was buried with a lock of his hair, his photograph, several of his letters and a ring that had belonged to Brown's mother, which he had given to her. Given the facts, I would definitely say there was a relationship there.. I just don't think it was a romantic relationship - and even if one or both of them had some romantic feeling, I seriously doubt that the Queen had it in her to physically betray Albert's memory.
I highly recommend the movie, and I think it was well-thought-out and respectful to both Victoria and John Brown. It portrays the depth of their friendship and understanding.