Hmm, Henry's proving to be an enigma. Out of curiosity I checked the following parishes to no avail, no mention of a Henry Cook/Cooke in the right time frame, either much too early or too late.
Ashburnham, Battle, Bexhill, Bodium, Brightling, Burwash, Crowhurst, Dallington, Etchingham, Ewhurst, Fairlight, Guestling, Hastings, Heathfield, Hurst Green, Hurstmonceaux (Herstmonceaux), Icklesham, Mayfield, Mountfield, Netherfield, Ninfield, Northiam, Rotherfield, Rye, Salehurst, Sedlescombe, St Leonards, Ticehurst, Wadhurst, Warbleton, Whatlington and Winchelsea.
This covers quite a large area around Watersfield but the area starts to go over into Kent for which I don't have access to records on-line. Anc..... only comes up with the Henry baptised at Westbourne but I don't believe that there are no other Henry's in the right time frame nearer than that.
There's also the possibility that he wasn't baptised or his baptism occurred in one of the few churches not yet transcribed on the SFHG site. Another possibility is that Henry wasn't the name he was baptised with or the first Christian name he was given, I've had numerous ancestors who've used another of their baptism names other than the first one.