So I have really bad Eczema / Dermatitis on my hands and I do my nails all the time, so it's fine in most cases you just need to be extra careful with products.
If the skin is broken near the nail and you can see it's open, steer clear. A lot of my skin breaks but nearer the nuckle so actually is fine around the nail. Don't get me wrong when I drip a bit of dehydrator down my finger it kills, but that's me doing my own nails. You'd need to make sure you don't over load your wipe with liquids to avoid it.
I use acrylic and then a gel polish. The product isn't going on the skin so it's fine.
The rebalance is where I have issues as the dust drys out my skin even more and can get really sore. So last week I put on some nytrol gloves and cut out the fingertips to ensure as little dust got on my hands and it worked a treat.
As I say, if anything is too close to the nail, side walls, finger tips you probably do need to avoid until the cracked skin is healed as if it's open it and can cause too make flare ups and infections.
The other thing is to make sure it's actually not an allergic reaction to the product that had been used. Sometimes the symptoms can look similar so always best getting a Dr to confirm for the client. You then at least know where you stand too.
I would also assume, that you won't be using the same product as an nss salon so that should hopefully solve the issue, depending on what the possible reaction was too.
Then when it comes to the end, ask whether they're ok with any oils or creams. I'm ok with the oil but not the cream!
Good luck