Hello Toby;
I thought I responded to this the other day. I don't see it, so it must have slipped between the cracks so to speak.
Assuming the old cutting edge is removed and the bucket is prepared to receive the new cutting edge, a "bridge" with a hydraulic jack can be used to bend the bucket back into alignment.
Measure the height of the jack in the collapsed position. Weld ears on the sides of the bucket extending down. Then, weld a channel or small beam across to support the jack. The jack will have to be strong enough to bow the bucket into alignment. I would straighten the bucket when the jack is released, so you will have to go past straight to get the plate to yield.
Once straightened, the bucket will bow again when you weld on the new cutting edge. Prebend the edge of the bucket maybe a 1/2 inch or so to allow for the bending that will result when welding the new edge. There are different shapes of the cutting edge, some go below the bucket and I've seen them placed along the top edge. You will have to determine which way to prebow the bucket based on when the cutting edge is placed (above or below).
The attached sketch is crude, but provides an idea of what I'm talking about. If you don't have a good jack, you can use wedges, just reduce the clearance between the "bridge" and the bucket.
Al