Horses were (and still are) very expensive prestigious animals. They eat a lot (eat like horses in fact) and I suspect our national trait of not eating them may also have been important. Oxen on the other hand are a part of the necessary livestock of a mixed farm - if you want more moo cows / milk / cheese you need at last one bull and mummy moo cow will inevitably produce a male or two during her calving. And you can happily barbecue / smoke / pickle / salt down the ox when it's too kernackered to pull a plough, or a cow.
As to when the swing to horse-pulled ploughs happened, I'm guessing mid 18th - early 19th century, when selective breeding of cattle and other stock led to better meat breeds and ?bigger carthorses. Anyone got info on the heritage of breeds such as the Suffolk Punch? That might give a firmer date to when ploughing habits changed.
My dad was taught to plough with 2 horses at the age of 10, he explained that horses were easier to use when fields became smaller and ploughs became smaller and lighter
Excellent link, jno. So by the end of the C13 the horse was being used substantially because its general versatility and speed of work outweighed the difference in 'running costs'. It was also much cheaper to obtain a usable working horse 'second hand', since an ox maintained its value for hide and meat but the horse did not.
Even so, the horse did not completely oust the ox until the C18, according to other sources..