The following possessions are still around today:-
Anne Boleyn - The Boleyn Cup was made for Anne Boleyn in 1535 when she was Queen, just a year before her death, and is made out of silver and gilt. It was apparently passed on to her daughter, Elizabeth I, who gave it as a gift to Richard Master, her physician, and then he eventually gave it to St John the Baptist Parish Church in Cirencester where it still lives today
Anne Boleyn - The silver gilt clock which was given to Anne Boleyn by Henry VIII. The clock features “DIEU ET MON DROIT” inscribed on one of the weights, as well as Anne’s motto “The Most Happye” and the initial H and A intertwined in a lover’s knot. The richly decorated clock also features fleurs de lys, miniature heads and a lion bearing the arms of England.
You can see a photo of it at http://images.library.yale.edu/strawberryhill/oneitem.asp?id=74
Elizabeth I - The locket ring which Elizabeth wore and which contained miniatures of her and her mother, Anne Boleyn, is kept at Chequers, the country residence of the British Prime Minister.
Elizabeth I - I think Hatfield House have some of her belongings including a pair of riding gloves.
Elizabeth I - A christening gown thought to have been worn by Elizabeth was found at Sudeley Castle and is on display there.
Elizabeth I - The Victoria and Albert Museum have a virginal once owned by Elizabeth - see http://www.vam.ac.uk/res_cons/conservation/journal/number_42/virginial/index.html