This fixes T68878
As shown in the report, during edit mode, when you have more than one
shapekey and you have the Shape Key Edit Mode option enabled,
changes to the Base Key don't result in any changes to the displayed mesh.
However, if you select another shape key, the mesh is updated.
Analyzing the code, the shapekeys update every time you convert a bmesh to mesh.
So the solution was to move the logic of updating shapekeys in
BM_mesh_bm_to_me to be used elsewhere.
But this solution makes use of a small COW hack by updating the key COWs
and copying the results into the original keys.