What you need to be considering re. dissemination of portraits are a) copies and b) printed versions: woodcuts and engravings, found in books, or by the later C16th, sold separately as picture prints. Both are more prevalent for Elizabeth, especially in the second half of her reign.

https://www.npg.org.uk/research/programmes/making-art-in-tudor-britain/case-studies/the-queens-likeness-portraits-of-elizabeth-i

Numerous other versions of this composition survive, including those at the National Maritime Museum and Hever Castle, which provides evidence of the increase in the production of the queen’s portrait in the 1580s.

Robert Tittler, 'Portraiture, Politics & Society' in A Companion to Tudor Britain ed. Robt. Tittler & Norman Jones (Blackwell, 2008)

Especially in England, and perhaps to a limited extent in Scotland, copies of portraits of eminent figures both English and (sometimes) foreign were widely made and sold from at least the half-way mark of the century. While some were painted by regional or itinerant craftsmen in provincial centres or great country houses, others, especially in London, were being produced by large workshops devoted to such work. Fruits of this production were commonly hung in the country houses of the day, in long galleries and other spaces alongside paintings of family and friends. They no doubt served as decoration, but also as declarations of admiration for or devotion to their subjects, and thus often as statements of personal loyalties in matters of politics and religion. By the mid-century it became increasingly common for such copies, in the form of prints as well as paintings, to be purchased and displayed by the middling sorts of people as well, so that even dwellings well below the level of the country house might have their pictures.(p.461)

Tittler's book Portraits, Painters, and Publics in Provincial England, 1540-1640 (Oxford University Press, 2012) on p.18 refers briefly to copies and woodcuts. p.159 refers to Home Counties nobles commissioning copies of portraits held in London. This book is on Google books. Have a look through it for keyword references such as copy, print etc.

If you were in the UK, especially England, it would be likely that local stately homes had the odd copy portrait of Elizabeth (usually the lower ranking the house, the cruder the artistic quality. Not a personal value judgement BTW, I really like those ones as I feel they give a stronger sense of a different time.) Not so many of those online, however. Various woodcuts and engravings of Elizabeth can be viewed online with image searches; some are from museum-type sources that provide year and place of production.