When I renewed my passport last year there were horrendous tales of long, long delays. I went to the main post office for our area - and asked them to do it. Simple as that. It cost me about another £10-£15, but it included them taking the photograph which I would otherwise have had to pay for.
I took my old passport and lots of proofs of identity and residence (none of which were needed, but after living in France I got into the habit).
They took an approved photo, which verified it, filled in the form on
computer, by asking me the questions and zapped it all off.
I got a new passport a fortnight later.
At the time, I repeat, there were huge delays, but this made it so easy for them to issue.
So it is worth trying. Good luck.