Can a UK passport holder obtain a 60-day tourist visa through a border bounce in Thailand?

You cannot obtain a visa by doing a border bounce. To obtain a tourist visa you would have to go to a Thai embassy or consulate in a nearby country and apply for a tourist visa. Some of them require an appointment and all of them require at least 1-2 business days processing time.

With a UK passport if you leave and return without a visa you'll receive a 30 day visa exempt stamp which you can extend one time for 30 days. Please note that you can enter Thailand via a land border only 2 times per calendar year without a visa, but by air there is no written limit.