Try www.allaboutorganics.co.uk - lots of local listings.
Or do a web search something like 'organic food delivery' - lots of stuff comes up.
I get meat every few months from Sheepdrove Farm (www.sheepdrove.co.uk or might be com). They deliver nationwide, but free for 'local' i.e. about 30-40 miles radius as long as you spend over £50-60 I think it is. Or sometimes I get a few meat extras added to my Abel & Cole order. We also buy a few bits from a local garden centre, although they focus more on free-range than organic. We have 2 huge freezers so tend to buy in bulk.
We do exactly what you say. We now only eat meat about twice a week (3 is a real luxury!), with fish once a week. So up to 4-5 days a week is veggie and we love it. It's much better to have really good-tasting, ethically-produced meat, but have it less often. And I'm sure it's better health-wise too.
I think our food bill has risen since going as organic as possible, but not drastically because some of our meals are so very cheap - we eat a lot of pulses and home-made soups and we also grow lots of veg ourselves.
We have decided that we are prepared to pay a bit extra for food because it's so important to us and we live modestly in other areas - e.g. we don't drink much, don't smoke, don't have cable or satellite TV, don't pay to rent DVDs etc, spend about £10 a YEAR on mobile phones. So while other people say organic food is too expensive, they spend loads more than us on other areas of their lives. It's all about choices isn't it, and this works for us.