I’ve heard that too but it’s hard to verify. Elizabeth I definitely bought a strands of pearls from the collection of Mary Queen of Scots after she was imprisoned but so much of Elizabeth’s jewels, dresses etc were broken down by the Stuart’s after her death. I remember reading somewhere that James I gave some to his daughter when she became Queen of Bohemia but she too was deposed. I’m currently working on an article about the Spencer Necklace which is supposed to feature some of these pearls.

As for the Cheapside Hoard, they are in the Museum of London but they are not on display. One or two pieces are on display in the British Museum but these are the enamelled chains

Shaun Leane did a fascinating documentary on the Hoard for the BBC and it is available on YouTube. Again, even the documentary states that it is hard to date the hoard but it could definitely be pre Civil War as they found a small carnelian signet stamp belonging to Viscount Stafford created in 1640