I agree with what
@Pa**
said. You can extend your 30 day visa exempt entry stamp by 30 days at the immigration office. That will give you almost 2 months.
During that time you need to apply for a new UK passport from VFS Global in the Trendy Building on Sukhumvit Soi 13. And as paul said it takes about 2 weeks right now.
getting the extension will get you stamped in long enough to get the new passport..