Great job! This is the time to be pruning those elms. Don't make the two elm pruning mistakes:
(1) If you want the branch to be thicker, don't touch it. Particularly if you are trying to get thickness into an apex (for example). If the branch is shading out the interior of the tree, remove foliage from the interior of the branch (leaf strip it) while leaving all the leaves on the end of the branch. Do NOT shorten the branch or cut the apical meristem (the end tip of the branch).
(2) If the branch is thick enough, cut back to two leaves. Not three, or four, or six. Cut all the way back.
This is the time to completely remove the branches you don't want. Given the time of year, don't forget to use waterproof cut paste on the wounds.
If people make any mistake, they tend to overprune the long growth they are supposed to keep, and underprune the tight ramification growth that they need to keep tight and ramified.
Thank you for the tips. I will certainly remember that, very useful. Similar advice given by Smoke, once upon a time.
However, I think with these 2 trees, there's not much to keep on them. I feel like the best thing to do would be to remove the majority of them and let them regrow and directional prune over next few years to rebuild a branch structure.
Looking at the pics (although difficult to see), do you think there is much to keep?
Although saying that... This doesn't look so bad. There is balance, the lower branch is thickest, next one slightly thinner so it has taper...