Disyllabic pronunciations of the word "medicine" are quite old, and have in fact at times been considered more proper than trisyllabic pronunciations (I mean "proper" in terms of the concept of "RP" or "Received Pronunciation", as Mick pointed out). It may be relevant that the modern French word, "médecine", is also often pronounced with only two syllables—that's just my speculation, though.
The Oxford English Dictionary (OED) entry for the word records spellings like "medcoyne", "medcyn" and "medsin" that it says occurred as early as Middle English, and includes the following note about the pronunciation:
The disyllabic pronunciation (recognized by Johnson 1755) has existed at least from the 14th cent., as the forms occasionally indicate. N.E.D. (1906) stated that the trisyllabic pronunciation was less common in England, but that in Scotland and in the United States it was apparently the prevailing usage, and that examples of it occur in verse of all periods, from the 14th cent. onwards. N.E.D. also stated that the trisyllabic pronunciation was by many objected to as either pedantic or vulgar. The trisyllabic form still predominates in Scotland and in North America; H. W. Fowler Mod. Eng. Usage (1926) recommends the disyllabic pronunciation, while subsequent editions note the increasing frequency of the trisyllabic pronunciation in England.
In the word "medicinal", the first "i" is always pronounced because it occurs in a stressed syllable, unlike in "medicine". But there is some evidence that in the past, some speakers used an alternative stress pattern for the word "medicinal" that resulted in the first "i" being unstressed and possibly left out. The OED entry for that word says
Johnson gives pronunciations with stress on the penultimate syllable as well as the antepenultimate. Walker adduces a number of authorities in favour of the latter and casts much doubt on the former. A trisyllabic pronunciation, with elision of the second syllable and probably with stress on the first syllable, is attested by 17th- and 18th-cent. spellings such as med'cinal, medcinal.
(You can see Walker's dictionary entry in the scanned copy of his Critical Pronouncing Dictionary (1791) on the Internet Archive.)
As far as I can tell, both pronunciation variants existed already before American English and British English split. The present-day difference doesn't seem to have any clear cause. By the way, the idea that "American pronunciation pre-dates a shift in sound of British English" is an over-simplification: both varieties have changed (although not necessarily by the same amount) from their common ancestor, like how humans and chimpanzees share a common ancestor that is identical to neither of them. (The “common ancestor” of American and British English wasn’t homogeneous.)