I'd give this 5 stars, except the instructions to use it aren't at all clear.

How I got it to work (to add google.co.uk as a default to google.com) in Windows Firefox v 126.0 64bit:

1) add (ie, install) the extension
2) restart FireFox
3) Right-click search box. You'll then be given the option to add a custom search; click this, then enter the details. Finish.
4) go to firefox->settings->search and your new custom search engine should appear at the bottom of the "Default Search Engine" drop-down list. Choose it.
5) Voila!

I don't like running extensions when not in use, so I turned it off. When I restarted FireFox, my choice (google.co.uk) was still active.

Thanks @Tom Schuster