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It's def more (£30-40, from memory), and you have to hang around for a few hours while they do their stuff.

Yep, you would (99% sure)... easy enough to do online though...

Yep...personally, I'd go for getting a new passport before travelling - you have enough time. .gov.uk says that you'll get it back within 3 weeks using their standard service (£72.50), or you can pay £103 for the 1-week fast track service (which also requires a visit to the passport office).