This is, like all history, a complicated question.

The concept of "available to the public" has always been a moving target. Even today, museologists are confronting the fact museums appear elitist and while open, are not truly welcoming to the public.

Today, the Tower of London is part of HRP (Historic Royal Palaces) which is a historic trust effectively run as a museum. Many of the objects inside (commonly called the Royal Collection) are held by the Royal Collection Trust and not the Queen herself. (Even the Queen's Gallery is overseen by the Trust)

The forerunner to museums were "cabinets of curiosities" - quite literally display cabinets of collected exotic materials owned by the wealthy. One example would be the Tradescant family was displaying their collection of "rarities" in their home and people could pay to see it in the 1630s. This collection was given to Elias Ashmole who gave this collection and his own to the University of Oxford in 1677 and stipulated that an institution be built around his collections for research and learning. This collection came to be known as the Ashmolean and the first modern museum. But in these years, the people who could afford it and the people who would be allowed to enter were scholars or upperclass. They were not open to the general public as we would define it today.

In 1699, the royal jewels went on display, and while I don't have information on *who* could see it, I would confidently guess that the people who were seeing it were fairly similar to those who were going to the cabinet of curiosities and Ashmolean at nearly the same years - be that the upperclass and the scholars (and the dubious connoisseurs who were upperclassman who fancied themselves as scholars). Furthermore, this list outlines a list of things one can see at "The Jewel House" (aka the Crown Jewels) - now, while a published guidebook implies that there was a "general public" who could go and visit (as opposed to invite only by the Queen), the sheer fact that the person touring could a. afford to tour and b. afford a book, means that they were already of some elevated status.

In summation, I've provided two answers, that seem disjointed, but somewhere in between there is the truth of the matter. We can establish when the jewels went on display, but this display would be a very different type of display than one would think of today. Even if the Ashmolean - opened a few years earlier - was considered the first public/modern museum, even that is still not how we define them today. I would posit that the world's fair - the first one being held in London in 1893 - would be the first time that a more modern and inclusive definition of public was allowed to enter and would thus be more "museum-like' by today's definition. But I don't know if the royal collection was displayed (I think it was more industrial arts).