Follow up question. Did veterinary care exist in that period?
John Clark has a book called The Medieval Horse and its Equipment c1150-1450 which is where I'll be pulling this information. The book is based on Findings from excavations in London so this will be relevant primarily to working horses rather than horses used in battle.
The first important thing to consider is that while horses were certainly a common in London most people did without.
For example when the king was moving his household in 1285-6 over 1000 horses were ferried across the Channel to provide transport for the royal party. The largest concentration of horses regularly seen in London was the horse fair held weekly at Smithfield.
However while horses were certainly a common sight most Londoners wouldn't have owned horses though they could be hired at considerable expense. Buying a riding horse would have cost something like 6 months to a years wages for a skilled London craftsmen and then come with an expensive upkeep. In 1389 there was an attempt to reduce the expense of the annual procession that accompanied the new new sheriffs to their presentation at Westminster one of the major expenses most objext to was the cost to 'the men of divers trades' of hiring horses to ride in the procession.
It's difficult to directly answer the question about how dangerous a horses life would have been but here are some details.
A modern horses functional life as a draught or riding animal is between 18 or 20 years of age. Heavily used, overworked, maltreated, and underfed animals would probably have a shorter functional life. The expenses of keeping a horse in a city like London probably meant that the horse would be unlikely to be kept beyond it's economic usefulness.
Horses are frequently aged based on having groups of incisors and in London there are two excavations that have found groups of incisors.
Using post crainal bone development most horse remains from medieval sites were adults older than 4 years of age and probably older than 7. Only two bones indicate animals less than 4 years of age out of the hundreds of bones that have been recovered and both of those were from a middle Saxon site.
None of the individual horse teeth that have been found show the level of tooth wear that would be expected on an animal of 18-20 years old.
Evidence of the cause of death is also very uncommon. One horse (from LUD82) shows evidence of it's death. That horse seems to have been pole-axed in the head causing a large depression in the top of the cranium. After being stunned from the pole-axe probably had it's throat cut or was decapitated.
A second skull from Ludgate Hill (LH74) shows evidence of decapitation wince it carries a chop mark across the basal part of the occiptal condyles where the head joins the neck.
Slaughter of horses at an early age suggests that maltreatment, accident, underfeeding and disease was probably common for horses without access to modern veterinary science.
Where sex could be determined in the horses (only 8 out of the hundreds found) all were male. This may indicate that mares would have been valuable for breeding purposes and kept largely outside of London. The unwanted geldings would have then been used as draught and riding animals.
TL;DR if you were a horse in Medieval London you were probably male, brought in as an adult, lived a hard relatively short life, and then were killed when you cost more to keep than you were worth.
Follow up question: how would horse survival rates compare to those of WW1?
Do we have any idea of what percentage of horses would likely fall in a battle? With arrows and spears raining down it always seemed to me like the poor horses had an awful time of it.