I think your confusion may lie in the structure and span of the UK's orders. I don't doubt, to keep this more widely focused, that many other countries will have taken the same approach.
There are, of course, many different vaccines being researched. The UK hedged its bets by placing provisional orders with a number of different developers, recognising that not all of them would ever come to fruition - the irony being that one of the two announced thus far, Moderna, was not one of those with pre-orders. Some of those options, with very large orders against them, have not really progressed at all and are likely to be overtaken by events.
So of that 335m outstanding, the reality is that perhaps around a third might end up actually being deliverable, and that over many months. That's enough, in time, to vaccinate pretty much the whole adult population but it doesn't leave much over.
In terms of the question being asked on access to private sources - in the UK at least that's not likely to happen, so that's one country off the tourism map!