Yeah theoretically you probably could, but reality the plane is just too heavy. You would need massive amounts of thrust to overcome the wings being useless, for an extremely heavy plane like the 747 the airframe would have be incredibly strong. As the plane climbs the engines become less and less powerful, and eventually they wouldn't produce enough power to keep the plane flying (and so it'll stall, spin etc).
Fighter jets can do vertical climbs because they produce massive amounts of thrust compared to their weight. But here's some fun facts for you. 1 747 engine produces almost the same thrust as the f-22's 2 engines do with the afterburner engaged. 1 777 engine can produce just under twice as much trust as the afterburning f-22. Those engines are ridiculously powerful as it is.
F-22 engine with afterburner: 35,000 lbs
747-400 Engine: 63,300 lbs (the 8l has slightly more powerful engines that produce 65,000)
777-300 engine: rated at 115,300 lbs of thrust, but has been tested at 128,000 or something like that.
The other thing is it would be an extremely unstable plane. The thrust would keep the plane in the air but the rear CG would keep forcing the nose up, and you'd have to have a massive horizontal stabilizer to keep forcing the nose back down. But having a massive horizontal stabilizer would make for a very very sensitive plane.
So anyways, yes a massive amount of thrust would theoretically work at first, but the other design considerations wouldn't make it effective. The only real way to fix such a massive load shift would be to well shift the load back to the appropriate CG range. If this plane was at altitude, they might have had the time to be able to do that. But they just took off and it happened so fast (they didn't even have the landing gear retracted) that they were pretty much dead the second the straps broke and the cargo fell backwards.
The most likely scenario was that the cargo the plane was carrying wasn't secured down properly, hence when the plane started climbing, the load shifted to the back and created a rear CG. If the rear CG was severe enough (like in this case) it'll cause the plane to stall and the elevators won't have enough authority to bring the nose back down. So the plane will remain in a stall, possibly spin and eventually crash to the ground (like in this case).
Generally speaking it's the captains responsibility to make sure the cargo he's carrying has been secured down properly and the weight and cg locations are within limits. After years though a lot of them get complacent and just trust that the load master did his job correctly (which they should have done).
The cargo that it was carrying was 5 MRAPs and apparently the final communication to ATC was a loud banging noise heard from the back which could possibly mean the straps snapped and the cargo fell backwards causing a cascading effect. Each of those MRAPs weighs 14+ tons, so that's going to be a major load shift.
