I have the Powershot A3300 (IS) which has this facility. It has 2 presets of 2 seconds and 10 seconds, then a third option where you can choose the delay to suit yourself, anything from 1 to 30 seconds. The same feature lets you specify how many shots it will take at the end of the delay you've set, anything from 1 to 10 shots. It's a super little camera, fitting easily into a cycling top pocket and the results are very good. You should be able to pick up one of these very cheaply, certainly under £40.
I recommend Canon's outlet on
Ebay where they sell refurbished cameras with a year's Canon warranty. You could get
this A3500 (IS) for £39.99
Anything I've bought from them has been indistinguishable from new.
GC