Welcome, good to see you stopping by that's what the forums are for!
I too have the BF1942 anthology set, but luckily I've had it installed for years without too many issues. I do recall having trouble getting it to launch a few years back after a patch or something but I can't remember how I fixed it. I'm going to talk to some people and get back to you, or maybe they'll come in and help you directly.
PS, just a random thought. Maybe try launching it as one of the expansion packs instead of the vanilla game? Sometimes doing funny things like that would get the game to work for me, or launching it by selecting the 'autorun' application located directly on the 1942 disk, via your HDD drive options in Windows Explorer.
Finally, the more system information we have the easier it is to help you, what OS are you using etc etc?