I doubt it.
I don't think there is such a thing as a Royal Coin Collection in Britain anymore - the remnants of any such would have been handed over to the British Museum ages ago, I suspect.
The Royal Mint probably has a collection of all the coins they've ever made, but that would only cover a small fraction of the total number of coins with the royal portrait. The Royal Australian Mint, Pert
h Mint,
Royal Canadian Mint and Pobjoy Mint would all have issued as many, if not more, coin designs with Elizabeth II on them.
The British Museum might come closer - since the 1800s they have strived for a "complete collection" of British coinage, and no doubt have a fairly extensive collection of Commonwealth coinage, too. The various national museums of Australia, Canada, New Zealand etc would likewise probably have complete collections of their own national coinages and incomplete collections of those from other countries.
There could well be private collectors striving for completeness, but unless they've got a very large budget, I suspect the goal will be out of their reach. And you'd have to debate whether the super-maxi-bullion "coins" of the
RCM and
Perth Mint count, since they've only made one or two of each.

Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis