Elizabeth I was dead when Jamestown was founded in 1607 (which explains it not being Elizabethtown, I suppose), putting anything found there ever-so-slightly post-period for this project. However, as I mentioned the other day, the lives of the common people changed very little just because there was a new butt on the throne.
In the kitchens especially, I am seeing the forms and functions of the cookware from the reign of Elizabeth to the reign of James I didn't change much except in materials.
Necessity being the mother of invention, when something already works there's no necessity to drive innovation. When changes were made, they were mostly changed based on differing climates and the availability of materials. For instance the half-timber "wattle & daub" methods of construction persisted in Europe well into the 18th century, but were set aside in America due to the ready availability of timber resources and substantial climate differences. Yet the ovens in the American colonies were almost identical to the ovens of old Europe.
Not to get too wonky on this, but the most obvious advances in technology between the late 16th and early 17th centuries were metallurgic. Advances in casting techniques and the efficiency of furnaces, however, suddenly made large-scale iron casting possible and cost-effective. James's rule saw the beginning of the cast iron age and the decline of bronze and brass as the go-to metals for every little thing. Foundries started turning out quality cast iron in the late 16th century, leading to things like bronze cookware being shouldered aside in favor of cheaper, easier to maintain cast iron.
Anyway, back to Jamestown. The preservationists and archeologists at the Jamestown site have a Youtube channel, documenting their discoveries and highlighting the murky time at the dawn of the European presence (for better or worse) on this continent.