BF5 Loadout selection buging out

Unless you have loadout mirroring turned on, then every weapon selection will only be for that faction and that combat role.
So if you select the MP40 for the British Field Medic, but then the next match you are the Americans, then it will select whatever weapon/combat role you used last as American, regardless of what you had selected previous game as the British.

Make sure that when you select the weapon you want for the Faction Class you want, that you do so for the combat role you want also.
A lot of times when people have issues with loadouts, it is due to confusion with Combat Roles.
Also, not sure if you play combined arms or not, but any changes you make there, does not carry over to the next match played, combined arms or multiplayer.

What I would do, to double check, is to go into your Company via the main menu and manually go through all 32 loadouts(4 factions, 4 classes, 2 role for each), to make sure each loadout is what you want equipped.

Hmm, that is very odd.
At a loss as what else to suggest for you to attempt.