John Upton:
The on board scrolling displays and the tube maps say it is spelt St John's
Wood with apostrophe but the platform roundels omit it.

"San John's Wood"? Interesting variation. :-)

So which is right?

Either, both, or neither, as you wish. There is no single definitive
source for the "true" name of an Underground station, and many stations
have had this sort of variation.

Of course, if the station was named after something, and *that* has an
official or universally used spelling, you might take that to be indicative...
--
Mark Brader, Toronto "Just because it's correct doesn't
make it right!" -- Jonas Schlein