The problem with relying on a written consititution is that it can easily be changed.
The USA is in a position where it seems to be perpetually stuck in a 50/50 split of the elected government, so it's not possible for major constitutional reform to be passed by one party.
On the other hand, the current government in the UK has a majority of 80 MPs - if there was a written constitution, it's likely they could simply pass a law to re-write it.
In the US you need a super majority (of legislators or states) to amend the constitution. You'd have to do something like that. But the UK government system isn't really set up like this, there aren't the checks-and-balances of the US. (It's just civil servants undemocratically dragging their feet/slowing down processes they disagree with... which is not democratic.)
The USA is in a position where it seems to be perpetually stuck in a 50/50 split of the elected government, so it's not possible for major constitutional reform to be passed by one party.
On the other hand, the current government in the UK has a majority of 80 MPs - if there was a written constitution, it's likely they could simply pass a law to re-write it.