They did use a form of embalming. Not as advanced but they would have barber-surgeons who multitasked because physicians thought they were too proper to do knifework. These would prepare a corpse with surgery; opening, disemboweling, washing, searing and anointing with preservatives, then wrap it in waxed linen cerecloth. This would keep it preserved for a while longer, though not as effective as modern techniques and didn't always work so well. If the procedure or the wrapping were done cheaply or improperly then the body might swell and even possibly burst. ("Elizabeth Southwell's Manuscript Account of the Death of Queen Elizabeth", Catherine Loomis, 1996). One spectacular account, and I do find it fantastical and polemical, describes Henry I's preparations thus:

"His corpse was carried to Rouen, where his bowels, with his brain and eyes, were deposited. The body being slashed by knives, and copiously sprinkled with salt was sown up in ox hides to prevent its effluvia, which so tainted the air as to be pestilential to the bystanders. Even the man who was hired by a large reward to sever the head with an axe and extract the brain, which was very offensive, died in consequence, although he wore a thick linen veil; so that his wages were dearly earned... The corpse being then carried to Caen, was deposited in the church where his father was interred; but notwithstanding the quantity of salt which had been used, and the folds of skins in which it was wrapped, so much foul matter continually exuded, that it was caught in vessels placed under the bier, in emptying which the attendants were affected with horror and faintings." ("Determinants of the revival of dissection of the human body in the Middle Ages", P. Prioreschi, 2001)

You can see an earlier variation on the same procedure; dissecting and salting with preservatives before wrapping. This was an expensive procedure most used for those whose bodies would be waiting for large-scale funerals and who might lie in state on display. But it was also seen as somewhat distasteful, given they would be cutting and burning your body. Therefore some people left orders not to do it, like Elizabeth I; though in her case there is a story that it was done anyway (which would make sense given it was waiting for funeral for a month; stinky Queen corpse equals bad.)

Specifically in regards to Henry V: "The body was embalmed preparatory to its conveyance to England, but so much honour was paid to the dead king's memory that the men of Paris and of Rouen offered large sums of money for his body to be buried in France, and such slow progress could be made by the funeral procession that it did not reach London until 11th November" ("On the Funeral Effigies of the Kings and Queens of England, with special reference to those in the Abbey Church of Westminster", W. H. St. John Hope, 1907)

Eleanor of Castile, that was Edward I’s first wife. Not only was her heart buried separately from her body, but so was her viscera. From the sources I’ve read, burying the body in three parts, as opposed to two parts (heart and body) was a pretty uncommon practice.