I have shot commercial Kynoch rounds in 375 X 2½ dating back to the mid-1930s with no problems. Just make sure you thoroughly clean your bore with wet patches to get rid of the corrosive primer residue after shooting. Your ammunition looks to be in good clean condition and should be OK to shoot.

Have we got a slight language mix-up? By "fat" do you mean "wax"? I've had 303 British ammunition with a wax coating. Either way, I'd wipe it off before I shot the ammunition just to reduce the thrust forces on the bolt head. Perhaps that's being overly cautious but I tend to be that way with older rifles. They're beautiful, mature ladies and should be treated with gentleness, care and respect and the results can be wonderful.