You have roughly two options. One is 'true PAYG', in which you topup by say £10 a time and the tariff is X pence/call, Yp/SMS, and Zp/MB. Here is a list of PAYG operators and their prices.
The other is that many of the UK 'traditional' networks have moved from true PAYG to a system where you pay £x for a 'bundle' of calls/SMS/GB that typically lasts for a month, but there is typically no requirement to keep paying - once the bundle expires (or if you exceed it) you end up on some default tariff, which will burn through credit fast unless you buy another 'bundle'. You topup in the same way, but the system may buy the 'bundles' automatically if you have available credit. Many of the carriers linked on the website above also offer such 'bundles', either as addons to their PAYG service or as their main offering. Typically the most use for 'bundles' is to get more data than you might get on say a 1p/MB PAYG tariff.
The next question is how to get hold of a SIM. You may find you can purchase a SIM on say Amazon or eBay and get it shipped to you (or to your accommodation ready for you to arrive). The carriers will ship locally but probably not internationally. If you try to buy a SIM at the airport / street where you're staying, you may find a limited selection - typically just the major networks, most of which now have punitive true PAYG rates. Sometimes family-run smaller stores (newsagents, corner grocery stores, petrol stations, etc) keep some SIM cards, although not a good selection.
Once you have the SIM, you can topup online or from your phone via a credit/debit card, at many shops (eg supermarkets - you ask them for an £X topup on XYZ Mobile), and at some ATMs (withdraw £X from your bank and it goes directly to your chosen mobile number). Some of the more obscure operators may be online/phone topup only rather than at physical stores. It may be more difficult to topup online with a non-UK card, depending on whatever fraud prevention measures are in place.