If you can stretch a little I have had a logitech M705 for years, hardly ever have to change the battery and works fine, they can be found for low 20s.
edit: seem to have gone up in price recently actually!
I'm a big fan of the Logitech Pebble M530 which scrape in just under £20 shipped. I've a few mice out for them. Lots of personal preference involved in mouse shape/function though!
Does anyone have a recommendation for a mouse under £20 shipped?
At this price level there's probably not much difference, but ideally something wireless and that I don't have to buy using Amazon.