Not a lot of written history before the mid 1800s.
There is a drawing dated about 1835 of 'Padstow Hobby Horse' which shows two musicians playing a fife and drum, several men in top hats firing guns into the air, a tall maypole in the background with someone trying to climb it, two or three people who might be dancing (or they could just be walking funny, with hands on hips), a man posturing in front of the horse - presumably the Teaser and an odd looking hobby horse. All the elements of the Oss are there but it looks to me as if it was drawn by someone who hadn't actually seen it but drew it from a verbal description. Because yes, it has big eyes but not those sort of eyes and yes, it has a big nose but not like that one. It does have a big red tongue but it doesn't stick out the way the drawing shows and the horses head on a stick out front is much more stylised in even the earliest photographs. The character in the Oss costume in the drawing must be well over 6 foot tall because he towers over all the other figures. Maybe it did look like that back then but I suspect not.
According to Donald Rawe in his book Padstow's Obby Oss and May Day Festivities:
"The earliest written reference to a hobby horse is actually in the old Cornish drama Bewnans Meriasek (The life of Meriasek) written in 1502. This at least points to the conclusion that hobby horsing was general to Cornwall at the time."
"Sir Walter Scott in his novel 'The Abbott' (1820) describes the Hobby Horse festivities at Lochleven, Scotland:
'Here one fellow with a horse's head painted before him, and a tail behind, and the whole covered with a long foot-cloth, which was supposed to hide the body of the animal, ambled, caracoled, pranced and plunged, as he performed the celebrated part of the hobbie-horse, so often alluded to in our ancient drama.'
A similar description of a hobby horse's 'frisking' is given in Hone's Every Day Book (1826), under May 1st."
Donald also mentions St Augustine preached against 'horse magic' back in the 4th Century so presumably it's true origins stretched way back before that and have been lost in the mists of antiquity.
Oss Oss
Robyn