I get the feeling that Tangobunny is more concerned with needing to empty his bowels rather than having a pee.
Never having been treated for prostate cancer but I have had to be treated for pelvic poisoning which resulted in horrendous bowel problems which went on for months so I appreciate the fear factor, especially of being in a stranger's car when the urge comes on you.
As Lyn says though, your bowel is likely to be mainly empty so all that is likely to come out is possibly mucus and wet "wind".
Pull on pants are available from Supermarkets and are not expensive and will give you the added confidence in somebody else's car.
I obtained a RADAR key (which I understand can be obtained from Age UK although I got mine direct.)
It is handy to have even when just out and about as when that urge comes you need to be able to deal with it without queueing for a loo.
Might not be so bad for a bloke perhaps but since all us women only have the choice of a cubicle waiting in line can be a major problem.
Good luck, I hope all goes well for you.
Sandra