I can't say for sheep specifically, but most immune response is targeted at initial infection with the parasite.

In human helminthiasis, the main defenders are T-lymphocytes and eosinophils. They produce a ton of cytokines that are mainly focused on damaging/inhibiting eggs or young, very small parasites during initial infection. B cells and antibodies may be involved, but--once again--antibodies are not going to have substantive effects on the type of large intestinal worms you are talking about.

While I'm sure the host body has some immune tricks to tackle adult helminths, and others can hopefully chime in with them, by the time you have a chronic infection with large worms the parasite-host relationship is pretty established. Many helminths produce immunomodulating compounds which actively tell the immune system to not fight the worm, and, for the most part, the host is now a host and spontaneous recovery without some sort of medical intervention is unlikely.

One way would be to simply look at the blood underneath a microscope. If you take a smear of the blood and then stain it with various dyes, blood cells would look different between different species.

The differences would be more extreme the less-related the species. For example, the red blood cells of reptiles and birds still have a nucleus and look like oval eyes. Human red cells, on the other hand, are just red circles-- slightly lighter in the middle than the sides.

Amongst mammals the differences might be a bit more subtle. For example, you might look for a specific type of white blood cell called a eosinophil. They aren't common but if you search the slide you'll probably find one eventually. Eosinophils have a lot of large granules in them that look pretty different from animal to animal.

There are, of course, many many other ways to identify the blood. Some are much more precise than this (genotyping being the obvious one.) But a peripheral smear is low-tech, inexpensive, and I think most fun.