Could a bad serpentine belt affect the alternator ?

You're going *** backwards with this. Since the serpentine belt drives the alternator, you need it installed to test the alternator output. The 12 volts you see is coming from just the battery. Do not run the car without that belt installed.

Obviously you need a new belt. Put the belt on it and see if the alternator works. Obviously if you are not spinning an alternator at all or even fast enough, it isn't going to work properly.

Assuming your shop knew what they were doing, if they are only getting 12v out of your alternator then it is cactus. For a charging system to work you need a functional alternator, tight belt, working reg/rec and solid battery. The alternator should be putting out around 18V - the extra output is useful when you have all the electrics operating at full tit during those winter nights. The regulator controls how much of that juice the battery is allowed to receive, as the battery prefers to run at 12.5V to 14V. Too much juice and it boils, too little and it goes flat. The belt drives the alternator. If the belt is loose it slips under load and the alternator doesn't spin fast enough to generate full power. If in doubt take the car to an auto sparky and he will tell you in five minutes where the problem is..

Does the alternator turn freely check the idler pulleys also usually I replace the idler pulleys when I change a serp belt. What maintenance did the manufacturer say to do obversely you haven't done that. If you alternator is only putting out 12 volt its most likely not working

If the belt is slipping, then it wont turn the alternator properly, and it wont put out the correct voltage.

Only real way to know is to fit the new belt, if the old one is worn out, then needs replacing no matter what. Once you have a new belt on, then you test the alternator again and see if it's working. Measure the voltage at the battery with a multimeter. With the engine running at a fast idle, it should read around 13.8 volts if the alternator is good. That means the battery is being recharged. If it only reads 12V then the car is running off the battery only, the alternator isn't working, and the battery will soon run down again.

Of course it's possible that you have a worn out belt, AND a bad alternator, or even a battery that's going bad, But the only way to know is to do some quick tests, with a good belt in place.