After poking around the TF2 wiki for a while I noticed that the crit rate listed for melee weapons says that the maximum they can reach is about 25%; that didn’t seem right to me, so I performed a quick test. First I did a control test, swinging at the air 100 times, and counting up the crit swings, I got 14/100, which is within tolerable levels for the expected 15% base, in other words, the achievement server I picked wasn’t running any crit mods. I then proceeded to kill bots with the heavy’s minigun, quickly getting my rolling 20 second total damage up to 800 (or higher) – so that I would have the max crit rate. I started to punch some bots while counting crits, but got a warning message of some kind, so I explained there was a point to what I was doing before restarting the test.
(9÷14 on that test, about 64.3%)
So I repeated it again, this time dolling out a full 100 hits, and counted 65 crits, making the max crit rate around 65%, and suggesting that melee crits scale more steeply than ranged.
The demo of all this can be found here.
Update: The wiki has since been altered, quoting this post as evidence! My quest for TF2 melee crit rate awareness is finally over!