Scan is going to show, even if you don't trust AH that Altitude is either climbing or level, ASI will be increasing if level or decreasing/stable if climbing and VSI will show a climb etc as well.
Yeah, with hindsight and from the armchair....IF the AH was the cause, in the soup, right after T/O, without a stby horizon, about to clean the airplane....good luck.
Many years ago I flew in the middle of the night (dead tired, although technically perfectly within FDR), on top, single hand from Thessaloniki to Stuttgart and the AH went very, very slowly banking the airplane on A/P to the left. It took me quite some time to figure it out and I was on top with some moonlight. A takeoff in fog is about the worst time to experience an instrumental failure, again, IF that was the issue on hand.