By way of reference, the football "marching bands" even in my region regularly play in temperatures down to about 35 F (2 C) or lower at least once or twice during a season with virtually no special treatment for the instruments.

Sitting on open bleachers through a typically 2 hour game, with a 5 or 10 minute parade at half time is a real endurance test for the players at those temperatures, which often shows a bit in the performance.

Reeds must be kept warm enough to be flexible, and flutists and brass players may prefer to stick the mouthpiece "someplace warm" just before starting to play, to avoid the "tongue on the pump handle" kind of problem.

(I wouldn't recommend subjecting a professional orchestra grade instrument to those conditions just on principle, if there's a choice; but I never heard of an instrument being harmed by it.)

With a metal instrument, there may be a rapid initial bit of condensation, but there's less with wood or plastic instruments. The first passage or so warms the instrument some, and the first film of moisture that forms seems to insulate the "fresh air flow" from the cold of the main tube of the instrument, especially on the non-metal instruments. After the first few notes, there may be additional condensation; but probably less than you'd expect, because the first bit of moisture acts as an insulator for the air going through.

With most brass instruments, there is a significant continuous flow of air through the instrument, so there's a fresh supply of moist air continually going through. (That's why they have "spit valves.") Flutes, especially, have a significant through flow only as notes start, and after that its the same air just vibrating inside with a fair percentage of it coming back out the blowhole, so much less "breath" actually goes through.

If you're playing more than one or two numbers, you may want to have a "swab stick" handy for a flute, to clear it after the first couple of tunes; but unless you have an instrument that's more than usually susceptible, at 50 F or above your only problem should be keeping yourself comfortable enough to play to your normal level of performance. Keep the fingers warm, and don't let the lip quiver.

Of course a rehearsal with conditions similar to what you expect (possibly including a simulation of how you will be dressed) will tell you whether you actually do need to make some adjustments. At 50 F, I'd expect the most likely adjustment would be a scarf or longer sleeves to keep you warm enough to play well.

John