I can share my wife's story about fighting eczema she got years before starting nail tech career, and which got worse as the result of doing nails. Actually, it's quite similar to yours. She started with doing her own nails to get some practice and didn't consider the allergies she had had by the time (it was low to moderate bumpy rash that only happened from time to time so no wonder she ignored it). Quite soon she realised it's becoming more serious so she started taking action as she really wanted to be in this profession.
Using thick enough gloves didn't really help in stopping the allergy but that's a must for sure. We bought some expensive ones but it was clear that wasn't going to reverse the issue. Liquid gloves even made the allergy worse. Then we started checking all the elements that might cause the allergy. She got herself a powerful nail dust collector, first with a bag then with a small HEPA filter. There were no improvements whatsoever, so I did some research and came to a conclusion that the chemical fumes from the gels (she doesn't do any acrylic nails btw) are the main cause and of course the dust collector with only HEPA filter doesn't help with those. The only solution to this issue (apart from stopping doing nails) was to buy a rather expensive device called "fume extractor" which is essentially a vacuum with a very sophisticated (depending on the price) filtering system. The carbon filter inside captures the chemicals and doesn't let them out. Most of the smells were completely or partially gone once she started using the extractor, so that was a good sign and gave us some hope.
It took her roughly two months of using the device to feel the difference and we think it really worked in our case. I'm not going to share any links as I'm not sure a humble new member like me won't be banned for that, but anyone interested is welcome to PM me.
A side note. If you have an allergy, usually it takes time for it to go even in case of no allergens coming in! How much time you will need depends on your immune system and how sensitive it is. So in our case it was only after two months of "reduced or minimum allergen intake" when we actually saw the difference.
In theory, a good ventilation system would also do and "carry away" all the contaminated air, but in practice not many salons have a really good one, while having one at home is an even more expensive option.