PAYGO (pay as you go) cell phones in London: where to obtain?

I think the £50 or £60 mark is where you're going to be for a phone with basic smartphone functionality. There's probably cheaper around with no internet.

Argos is a good bet. They will have stock, hopefully of the phone you want. Be aware that it's usually just Tesco Extra stores (the large supermarkets) that will have stock of phones. Tesco Express or Tesco Metro don't stock phones. You would probably need to travel to your nearest Tesco Extra from central London. There's an Argos in Tottenham Court Road.

There's mobile phone stores in every neighbourhood in London. I can't see them selling a phone for much less than the £50 mark, though you could shop around. The brand is fairly inconsequential at that price, it will probably be Chinese brands you have never heard of. Cash Converters stores are common if you want to look for a used phone. It might be an idea to shop for a used iPhone or Samsung on eBay or Craigslist, as long as it's unlocked, before you leave the US.

Every corner shop (convenience store) will be able to sell you a SIM card and a PAYG voucher for whatever bundle you choose.

Lyca and Lebara are probably the cheapest PAYG SIMs around.

I looked into buying a cheap little unlocked phone on A'zon before my trip but didn't pursue it, I'd suggest doing either that or E'bay beforehand just so you have it. Make sure it's a quad band unlocked phone.

You may also want to peruse swappa dot com to see what reasonably priced used smartphones are available. I've both bought and sold tech devices on that site with good success.

I didn't mean to sound like I was denigrating Chinese phones in my previous post. It's just that down in this price range you're not seeing the best.

If you were thinking of picking up a phone before your trip, there's some real bargains to be had from the Chinese manufacturers. If you're willing to ship from sellers in mainland China, there's good phones at reasonable prices on sites like Alibaba.

If you're savvy and interested enough to delve into tech specs, looking at what processor, what version of Android it ships with and what screen it has is going to tell you quite a lot about how usable a low price phone will be. The Chinese made phones are more likely to have dual SIM slots, which may or may not work for you with your Sprint SIM and a throwaway UK SIM.

I had a look at EE on your behalf and they have some cheap-ish PAYG phones. The no contract "subscriptions" for £15 or £20 a month might work. You'd just need to check they aren't locked to EE if you wanted to put a super cheap Lyca or Lebara SIM in. EE have stores all across the UK.
https://ee.co.uk/mobile/pay-as-you-go-phones-gallery

just remember, given the huge trade in stolen phones in London, that if it looks better than it should at your price range it is probably hot.

When buying a used phone you also run the risk of something dodgy being left on the phone which might track your activity more than you'd like.

I would try it out in the shop with a working sim. I bought an Amazon phone on a holiday after mine got wet and didn't work. It looked good on the surface, with good specs, but it was so slow as to be useless except for phone calls.

Argos is your place,as above.
If you decide to try a indy phone shop, I have had oneplus phones for years, unbeatable bang for the book.