I find with most licenses, logbook, tax related issues its often best to just go to the DVLA fill out the necessary paperwork there and then and hand it over to someone you can acknowledge by name, that way you've got the estimate from the horses mouth as it were.
I recently changed address twice (moved house and it had sever damp damage/plumbing problems) anyway I had to change my log book and driving licence details, first time I sent them off took over a month (in Northern Ireland btw duno if thats any better or worse) second time I took them to the belfast office (also needed a tax disc) everything arrived within the week, so yeah lol.
unfortunately though most tax offices are in major cities/locations and so forth so it mightent be the best course of action
Sure you've sent off all the relevant things? Photo etc, mine took about a week for them to send me a form to send a photo with as I didn't want my passport one to be used, then 2-3 weeks after I sent that I got it. Something like that anyway, but I applied for it as soon as I could so it was there when I turned 17.