for milk intolerence of a more severe nature, any milk from an udder is going to cause a problem.
goats milk is only slightly different from cows. its a mammal thing!
from the nappies you discribe your baby COULD (but may not) have a milk intolerence, when the protien irriates the valve that sits at the top of the stomach. this swells and doesnt sit well, and so lets out milk causing reflux. due to the ratio of stomach acid to food being incorrect, food is only partially digested as it goes through the gut. Which then inflames your gut/tube lining.
So, go to GP , perhaps with one of these nappies?, ask for a refferal, to a pead, and a dietian. if there is a problem, IT NEEDS A MEDICAL PREFESSIONAL TO SORT THIS OUT.
ask gp about going totally milk free and what safe substitutes you can use, (bear in mind that some with severe milk problems MAY have problems with soya milk,so be aware of that.)
due to gut being inflamed will take some time for gut to heal and all symptoms to improve, so 6-7 weeks for some babies.
due to inflamed gut may be dehyrated so plenty of water to drink.
may have gut pain when laid flat in bed and so have problems with sleep, so prop up at one end if possible.
hope this is useful and you get more support from your gp.