Manual gearboxes generally work better/more smoothly when they have some heat in them and the oil gets a little thinner and better able to lubricate. So you’re right on that account. If the gearbox was crunching on every gearchange, you’d also probably suspect the clutch might be worn, too. But since it’s only the one gearchange that’s making the noise, then your theory that the synchromesh rings are worn holds water.
But before you do anything drastic, make absolutely sure the clutch is disengaging fully and that all the clutch lines and pedal action are spot on. A dodgy clutch can also produce this sort of crunching noise. And, yes, it can be worse when the vehicle is cold.
Meanwhile, for some reason, the synchromesh on second gear is the one most likely to fail on the majority of cars (probably because second gear cops and awful hiding and copes with very high loads). So, in that sense, yours is a copy-book case of worn synchros. (First gear is also highly loaded, but since you only really shift up from first – hardly ever down to first on the move – its synchromesh rings don’t cop such a spanking.)
If the problem gets worse, you’ll probably find the upshift from first to second will also start to graunch. That’s when you’ll know for sure that the synchros need replacing. As you point out, though, to really diagnose this the fault requires an inspection of the parts and that means the gearbox out and pulled apart. While you’re in there, change the other synchromesh rings and the transmission’s bearings. And don’t put it back together without a new clutch.
In the meantime, however, if you’re prepared to shift very slowly and carefully while the vehicle is cold, you can actually drive around this problem. Many car owners have done so for years without further problems. Not best engineering practice, perhaps, but it’s a money saver.
