Welcome aboard Flea! Always nice to see UK posters.

Could you clarify a bit? You say you're in the Big ring going UP a hill? Are you talking about the front chain rings? If you are, the chain slip problem apart, shouldn't you be on the smaller rings when you're grinding up a hill? And you sure got some hills in Wales!

Coz the big chain ring with a big rear gear is forcing the chain to have too much lateral movement (see EXCELLENT diag attached!!!), and may encourage it to jump off

Chain stretch should be easy to measure - each link is an inch. measure 12 links against a good steel ruler, should be spot on 12 inches. Someone (D*Alex?) will tell us what %age stretch is acceptable ie 12 1/8 inches=OK 12 1/4 inches=too stretched.
(Ah - I've just seen Steele-Bike has beaten me to it!!)

Another possibility mat be that the chain ring is warped/bent. With the chain off, hold a pointer near the teeth, braced against the frame and spin the pedals, should quickly show you if the ring is straight or not.

How old is the bike and its drive train? the rings & rear mech do get worn and break teeth. May need replacing. If they do, you can consider changing the cog sizes to hel with them Welsh hills, and may as well replace the chain at the same time anyway.

Stew