I'd be a little weary on relying on 3UK coverage - while it's generally grand in well built up places like the south of England, looking at 3's coverage (which is marked out as "Mobile Broadband" coverage) around the Lake District it's fine around the main towns and major roads but not that great off it, which will depend on what you're planning to do. AFAIK 3IRL PAYG sims will only roam on to 3UK and no other UK network. If you picked up a 3UK PAYG sim, the 2G fallback to Orange has been switched off in some places and where it's still available, data access is more or less crippled to the pages of a reduced Planet 3 service.
Vodafone, O2 and Orange/T-Mobile OTOH have better 2G/3G coverage than 3's 3G coverage alone in that area. With an Orange or T-Mobile sim card, you can roam on to each other's network at no charge either, 2G should definitely work and 3G probably will as well. Also T-Mobile should have 3G coverage where 3 has anyway. 2G is better than nothing and Voda, O2 and Orange have varying levels of EDGE coverage alongside GPRS.
A PAYG sim is easy enough to get. If you know the place you are staying first will take post addressed to you, you could go on to any of the operator's web sites and order a free PAYG sim card to be sent to that address. Failing that, many newsagents, small shops and supermarkets will have the same sim cards on sale for about £1 or less.
As for data, both Vodafone and Orange have a web PAYG bundle available of 250MB for £5 which lasts 30 days while O2 has 100MB for £3 or 500MB for £6 with free unlimited BT Openzone WiFi for the latter, again both lasting a month, although O2's 3G coverage doesn't appear to be as good as that of the other networks albeit that they also have 3G operating on 900MHz as well operating alongside 2100MHz on most sites which if your handset supports it gives better in-building coverage and slightly better 3G reception in fringe areas.