I had this fiasco to deal with a couple of months ago. My license expired 2 weeks before I got the replacement, thanks to the wife not sending my envelope recorded delivery when I asked her to post it!! My employer is mega ■■■■ and wouldnt let me drive under S88 as I had no proof of sending it. I ended up having to take holiday and a couple of days unpaid.
From sending it by normal post, to receiving it back took 6 weeks.
Regarding the medical, I used motormedicals. Cost me £49. Was spot on.
GP quoted me £150!! Ive used GP a long time ago and they didnt do any more tests than the cheap one did. BP, height, weight, brief eye test, and asking all the questions off the form. Id never use GP for it again.
(employer paid for medical, I refused to use GP on principal, plus the GP couldnt fit me in for another 2 weeks which would have been a disaster in the circumstances!)
Id be amazed if a HGV medical would uncover any underlying health issues that you couldnt work out yourself by buying a £20 BP monitor or a pair of scales!