I intend submitting a UK settlement visa application for my wife next month. Her Thai passport expires in August, and I would like to renew it before I submit the application. However, renewals usually take 1 month, and they're currently taking longer due to the protests in Bangkok.

Could I apply for the settlement visa using her current passport, and then obtain a renewal after the settlement visa has been issued?

Also, would the settlement visa need to be transferred into the new passport, or would my wife be able to show both passports when she enters the UK?

Yes, if she is unable to renew her passport then she can submit the application using her current passport.

When she renews her passport she will get the old one back. She can either pay to have the visa transferred to her new passport or do what most people do and carry both old and new passports.

The visa will still be valid even if the passport it is in has expired.