You can fast-track a passport renewal application - it costs £112.50 for the one-week service. Seems a lot, but the normal 4- to 6-week service costs £77.50 so it's not that much difference, I guess.
There's a same-day "Premium" service too but I just checked the directgov website for the cost and it says the Premium service isn't available for renewals, only new passports. I don't think that's true though: the only reason I know about any of this is that a friend recently posted on Facebook that she'd arrived at the airport for a trip to Bali and been refused boarding because her passport had less than 6 months left on it. Someone advised her to re-book her flight for the following afternoon and get an emergency renewal appointment the next morning, which she was able to do.
If you've not actually booked yet, though, I'd go for the one-week option and then at least you still get a holiday this month.