Learn English – Meaning of: “No man is an island, entire of itself”

Phrases.org.uk concurs with the OP that tide referred to a period of time:

The notion of 'tide' being beyond man's control brings up images of the King Canute story. He demonstrated to his courtiers the limits of a king's power by failing to make the sea obey his command. That literal interpretation of 'tide' in 'time and tide' is what is now usually understood, but wasn't what was meant in the original version of the expression. 'Tide' didn't refer to the contemporary meaning of the word, i.e. the rising and falling of the sea, but to a period of time. When this phrase was coined tide meant a season, or a time, or a while. The word is still with us in that sense in 'good tidings', which refers to a good event or occasion and whitsuntide, noontide etc.

Wiktionary confirms this in its etymology for the proverb:

A figura etymologica – time and tide respectively derive from the Germanic *tīma- and *tīði-, which are ultimately related.

A figura etymologica (or etymological figure)

is a rhetorical figure in which words with the same etymological derivation are used adjacently. In the narrowest definition, it is restricted to specialized uses of the accusative with cognate verbs (e.g. live a good life, sing a long song, die a quiet death). In modern linguistics, this same construction goes by the name of "cognate object construction" abbreviated COC. In its less restricted sense, the figura etymologica refers to just about any sort of repetition of cognate words in relatively close proximity to each other.

Its use in rhetoric suggests that it's simply window dressing for additional emphasis.

Related terms: polyptoton, hendiadys.

While I initially interpreted the phrase with a modern understanding of "care", as in to tend to, I did a check on the etymology of "care" to see if it could have possibly have another meaning and this is what I found on Etymonline.com:

care (n.) Old English caru, cearu "sorrow, anxiety, grief," also "burdens of mind; serious mental attention," from Proto-Germanic *karo (cf. Old Saxon kara "sorrow;" Old High German chara "wail, lament;" Gothic kara "sorrow, trouble, care;" German Karfreitag "Good Friday"), from PIE root *gar- "cry out, call, scream" (cf. Irish gairm "shout, cry, call;" see garrulous).

Based on the interpretation of care to mean anxiety, burdens of mind, I believe the 2nd half of the quote is saying that the medicine, though it "cares" for the wound, also brings worry and concern when applied too late.