The term 'bonsai' just means a particular way of training and pruning a plant, any plant that's suitable. I'm not sure what your plants are, need a closer, much clearer image of the leaves, but its clear that someone has been using some bonsai pruning techniques on them over some years, possibly even an attempt at cloud pruning at some point.

I like the one in the bottom photo. It would make a nice focal point if you cleared away everything else around it, leaving it standing alone. Then add some beautiful companion plants for it, irises, lilies, endless choices. The top one can go! :)

They look like pruned dwarf conifers:
top pic might be: Chamaecyparis obtusa 'Nana gracilis'.
bottom pic might be: Chamaecyparis obtusa ‘Pygmaea’

They could easily become bonsais. If you have a local Bonsai Society, they could give you instructions on how to save them--and/or will make offers to take them off your hands, Mandolinwind! :D

Looking closely at your photos it seems that these trees are set between hardscape material which means that the roots of these old trees are forever intertwined with this material. To attempt to remove them would permanently damage their roots ending in their demise.

The Chinese and Japanese have been collecting wild trees on mountain tops and cliffs, and turning them into bonsai, for centuries. In fact, it is the earliest form of the art. It takes a certain amount of skill, but there are techniques for that. Check this out:
http://kentbonsaiclub.weebly.com/

Agree the first one is probably be nana gracilis though its pruning has demolished any "gracilis"! The other one is a lovely shape and I would certainly keep it.

These are not Bonsais. A Bonsai is a potted plant trained in miniature. These are topiaries.

I agree that these are not true bonsais, YET, Bathgate. Right now, they are neglected shrubs more than anything else, though topiary may have been the original intent. My point is that many bonsai artists would drool over the opportunity to take these hardcases, and turn them into living works of art. The Kent Bonsai Club, or similar organizations, would be a good plae to find advice on how to do this, or people willing to adopt them, if they don't fit into Mandolinwind's plans.

That's true, the two crafts are similar. Just about any plant can become a bonsai or topiary, except maybe a saguaro cactus. I just like the twisted gnarly bark in the 2nd photo. I think it could be a focal point.