Just starting playing the first part (not lost company) and can't change weapons. I'm in the trenches, picked up the shotgun, rifle and Colt 1911. I have the 1911 and dynamite but can't switch away, even though I have ammo for the other weapons. I have the mousewheel bound to changing weapons but it's not doing anything. I tried selecting / deselcting the weapon menu in option and that didn't do anything. Pressing the numbers just shows images of the weapons, pressing fire doesn't select them.
Windows 7
6gb ram
ATI Radeon 4gb
Realtek speakers

I have the same problem, nothing works. Just started game and am where I need to shoot down planes using machine gun. I have set E to be the Use key and I receive a message to press E to take control of the machine gun. Pressing E does not work.
No matter what I do I cannot change weapons.

Weird. Have either of you looked at the game's config.ini to see if the binds are being saved or if they're just blank?

GOG's forum software won't let me post the entirety of the config.ini into my response, so I've tossed it up here - https://dumptext.com/SR09bqU4

Possibly copy its contents and replace your own config.ini to see if that might help. A word of warning, however, that all those settings are defaults. If you've got other settings already in place tailored to your personal experience, just look for:

Cfg.KeyPrimaryNextWeapon = "Mouse Wheel Forward"
Cfg.KeyPrimaryPreviousWeapon = "Mouse Wheel Back"
Cfg.KeyPrimaryUse = "E"

If those are blank, just replace them. If they aren't in it, add them in.

My ini file shows the next/previous weapon and Primary Use key as you stated.

A workround I have found which more or less works is to use a cheat for all weapons, ie. Press tilde (just above Tab) and type nvnweapons.
This gives what seems to be all of the weapons, when the weapon switching will work.

It's honestly strange that it doesn't behave like it should. It had been years since I last played the game and only re-installed it to see if it was a recent-ish problem and found that I could use the mouse wheel as intended.

But, if you're good with using the console command to give you the weapons then I guess it's not a big deal. I'd argue that it gives an unfair advantage by letting you have the Shadow Hand accessible before you're intended to get it (not unlike getting the Soul Cube in Doom 3 before you get sent to Hell, etc.) but it's also unfair of the game in general to not have a basic ability such as weapon switching being in a workable state so it balances itself out.

I was thinking about this more, but can't test out the theory I have since the game works the way it should for me.

But the idea I had was to potentially delete the config.ini and force the game to recreate it on the next run. It'd probably do nothing (and it's definitely worth making a copy of the file before doing away with it in the event it doesn't fix anything) though.