Well that is exactly what happend on a train I was on earlier this year.
The FCC train I was on had just gone through Sandy, when the train came to a very fast stop. At that location, there is nothing but fields, and too add to the problem, we were on the fast line.
The driver announced that a passenger had been taken ill, and asked if there were any medical people on the train who could help to go to carriage x.
After about 20 minutes or so (with a couple more requests for any medical people), we set off again, and the driver announced that the train would still call at St Neots, before carrying on to Huntingdon where an ambulance would be waiting (presumably because St Neots only has access to the platforms via stairs, and at Huntingdon the hospital is only a stones throw from the station).
It did seem a bit odd to stop somewhere where it would have been very hard to get access to the train, and although the decision that was eventually taken was the sensible one, it did seem to take a long time to get there. I would have assumed that the train companies would have had a standard set of procedures - passenger taken ill between x & y, do z, between a & b, do c, etc.