I am riding the south downs way next weekend for the first time – can I ride it on a CX bike or do I need an MTB (SS)?

CX bike will be fine 🙂 although bumpy 🙂

Cross bike is fine.

I'd rather do it SS mtb than CX but you can ride it on a CX bike as long as you have a high pain threshold and are prepared to wait for your mates on MTB's to catch up on a regular basis 😉

Done it on both.
Rode the SS 29er from B'ton to Winchester yesterday, and had to push a few hills.
Cx bike will be ok, you will have to take it steady on few descents and tougher tyres may help, as the flints can slash light cx tyres.
J.

Wear two pairs of gloves if you do it on cx bike.
J.