South East, Hampshire: the following are fairly near to you:
Wimbledon Greyhound Welfare/Hersham Hounds Their kennels are in KT12, they have volunteers who come daily to walk the dogs, lots of adopters send their dogs to the kennels when going on holiday. Looks like a good bunch of supporters. They seem to know their dogs very well and could easily help with high/low prey drive and all those issues.
Holbrook Animal Rescue As well as rehoming local dogs, they also work with Zante Strays and Desperate Greekies and advertise their dogs on the website. If you wanted a puppy, there are always a lot of lab/pointer type mixes from Greece/Cyprus. I know this is a massive generalisation, but Greekies have a great reputation for being lovely, easy dogs. There are a lot of poodle type crosses in Cyrus/Greece, exactly like cockerpoos here!
Desperate Greekies They also have an FB group you could join. Rehome mainly in the SE. You could also offer to foster for them.
Diana Brimblecombe Animal Rescue A bit further away in Berkshire but worth keeping an eye on.
Silver Fox Dog Rescue Rehome all over the UK. Can also offer to foster.
Poodles in Need If you like a cockerpoo, why not go for a poodle? Proper dogs, if you don't have the daft show-type clips. They have a lovely 8 month old (ref AV296) black miniature poodle at the moment on their website.
I would say that you have to be fairly proactive if you want to consider a rescue. Join all the relevant FB groups as they're always more up to date than websites. Also, "popular" family type dogs get rehomed much more quickly so often don't even get to the website, so you'll see these more on the FB groups. You can also get a feeling for the rescue, see how they react to any problems adopters post about etc. Make contact, get homecheck(s) done, keep in touch, don't expect them to ring you, they are so busy and lacking in manpower that they simply don't have time, so you have to make the effort. I know it seems a lot of work but, if you've got a bit of time, you will save 2 dogs (space for another one at the shelter) and have saved £100s that you would have shelled out for a pedigree.
You can also look here for other rescues. I would avoid the one which has a name like the liquorice "many varieties" sweets as I've heard they don't have a great reputation.