It has nothing to do with the embassy, applications are received at VFS offices in Bangkok and Chiang Mai. Look here for all the information needed;
Overseas British passport applications – Apply for a passport – GOV.UK
It is highly unlikely you will get your new passport by October, currently they estimate 11 weeks. You may need a trip to the UK to get sorted, or perhaps leave and re-enter on your Irish passport and do the same when the new British one shows up.