A drive belt that squeals only for the first few minutes after starting the car, often indicates a faulty battery.
Operating the starter motor on a cold engine and in cold weather can draw over 500 Amps from the battery (diesel engines over 700 Amps). Once the engine has started, the alternator will sense the drop and work overtime trying to recharge the battery. In doing so, the magnetic field in the alternator can increase to such an extent that it acts like a brake on the alternator pulley and the belt will slip. After a few minutes the battery will have recovered enough for the magnetic field to reduce and, therefore, for the squealing to stop.
This doesn't happen so much in warm weather because when the engine is warmer the starter motor needs less current to turn the engine and the battery (being a chemical reaction) is also more efficient when warmer. The mechanic who said "to leave it and see how it goes, and possibly the warmer weather might help" would be correct but that doesn't cure the problem.
I've come across numerous people that have kept on tightening the drive belt trying to stop it squealing without success. Fit a new battery with the correct CCA (Cold Cranking Amps) for the car and the squealing stops.
If the drive belt is set to the correct tension, next thing to do is get your battery tested under load. Don't continue to tighten the belt or eventualy the alternator, water pump, A/C compressor, PAS pump, belt tensioner, idler pulley etc. (if fitted) bearings will be damaged.
Dave.