Can I enter Thailand on a 60-day visa exemption, go to the Philippines, and then return for another 60 days with an extension?

Yes, as a UK citizen, you can enter Thailand on a 60-day visa exemption, leave for the Philippines, and then return to Thailand for another 60 days. However, note that you can only obtain a 30-day extension on the 60-day visa exemption when you return, not a 40-day extension as you mentioned. It's also essential to ensure your passport does not show excessive visa-exempt entries in the past year.