Simply look for any phone which meets your requirements and is advertised as 'SIM-free' (or similar), then get a SIM card from Asda, GiffGaff or whichever cheap provider you prefer.
I bought an Alcatel A3 earlier this year and I've nothing but praise for it. Most retailers were selling it for around £130 but I got it from Argos for £90. They reduced it to £70 later in the year but have now discontinued it. However the Ideal World shopping channel are still offering it for that price, with free delivery:
https://www.idealworld.tv/gb/pp/alcatel-a3-5inch-hd-android-smartphone-13mp-16gb-piano-black-457708
(It's worth three or four times that price in my opinion).
For alternative phones I suggest avoiding specialist mobile phone shops and checking out what you can find in Argos or the major supermarkets. (Remember that you're looking for a SIM-free phone though. Most retailers offer both SIM-free phones and ones which are locked to specific phone service providers; you should definitely avoid the latter).
I put an Asda SIM in my phone. (You can order one online for free or pay just 50p in store). I'm currently paying £12 per month but I can probably reduce that to £10 per month. (Unlike a contract, you're not fixed to a certain payment. You can change it every month if you want to):
https://mobile.asda.com/bundles
I can recommend Asda Mobile, which uses the EE network, for their excellent customer service. Calls to their help lines get answered instantly, by friendly and knowledgeable staff based in the UK. The only provider I know that seems to get just as much praise from other ABers is GiffGaff, so they might be worth considering as well:
https://www.giffgaff.com/sim-only-plans