The information in this link is out of date. The re-entry permit counters at BKK and DMK are both 24 hr now. The office at BKK used to have patchy hours but since they moved the service past security screening it's always open .
You should contact VFS - the British Consular services (including new passports) have been predominately outsourced to them now) - they should be able to advise you about getting an emergency / temporary passport. The turnaround time for new passports seems to be a around a fortnight.
Hi Laura - I did the Singapore online type B application last month - I entered the flight that I thought I was going to return on (hadn't booked it yet) but ended up coming back later (same airline, different flight). No issues or questions. They also didn't ask to see my booking when I was at the Embassy.