Unfortunately the UK apparently does not do LoTW.
I think if you check into it, you'll find this isn't strictly true.
The world is full of hidden restrictions (on money movement, on mail) to and from certain countries, but as long as there is a functional internet, an LOTW account should be possible. The only way it wouldn't be possible is that whatever convoluted stuff ARRL requires for documentation from rarer places is stymied by mail restrictions when relevant.
But, I don't think Uzbekistan is currently on anyone's restricted list. There's only a handful of operators DXing from there and it could be they have all elected not to participate in LOTW. There are several places in the world (several pretty active Southern Atlantic countries, for instance) where LOTW is fairly rare, probably because they are remote enough for the internet to still be a hassle.
And, while few now do so, it is possible to get DXCC and other ARRL awards strictly with cards, still.