For the cost of a Pico or a Zero? Even some of the SBCs cost about £30. Where can I get a mini PC for that?

ebay :-)

https://www.ebay.com/itm/147663241534

I'll bite. Wyse 3040. $20-40 on the used market, similarly crappy specs to a Zero.

https://www.parkytowers.me.uk/thin/wyse/3040/

Atom x5 Z-8350, 2GB of RAM, 8-16GB of flash, some USB, and there's a board on that page you can have done up to do a slapdash SD adapter inside the case for more flash.