I think you may be ok, but it would depend on the TOC policy. You've been offered the job subject to passing I presume?
Some people have been known to fall down on blood pressure because they are anxious about the medical, or even because they were rushing to get to the appointment. It would obviously depend on the underlying cause, what did BUPA say at the time? Have they referred you to your GP?
I would visit your GP ASAP and start by having your pressure taken again. If you still have problems then they will obviously investigate the problem, but if it is satisfactory then you may simply have had a 'bad day' at the medical. I would contact Southern and ask them how they wish to proceed, particularly if your GP reads your blood pressure as being OK. Southern may simply send you back for another try.
Best of luck.