It is actually extremely common for a celiac individual to experience more food intolerances and more digestive issues after going gluten-free, at least for the first several months (or up to a year). Celiac boards are full of anecdotes about this, and it was my experience as well. Very frustrating when you're hoping for a quick recovery, but it might not be a bad sign. A weakened immune system that is constantly being exposed to gluten will not react as much in any given case as a system that is further along in its healing.

I agree with Eugenia that a still very damaged gut can lead to the body sensing and "complaining" about improperly digested proteins, and that it takes time and a lot of tweaking to heal and repopulate a newly-diagnosed celiac gut. (Probiotics are a good idea). But I also think these flares happen a lot on a new gluten-free diet because, as you know, it's an autoimmune disease, not a simple allergy. With celiac, the immune system is triggering inflammatory reactions in the small intestine. Autoimmune diseases are very tricky to heal. It takes a long time for the body to stop reacting to perceived threats. Some people need to eat gluten free for 2 years before they can eat dairy again. A gut-healing protocol might help but I think patience is really important too. Good luck to both of you.

"Lactose intolerance is frequently a side effect of celiac disease. Celiacs who eat gluten become lactose intolerant after the villi and microvilli in their small intestine become damaged, and are no longer capable of catching and breaking down the lactose molecule.

The problem usually disappears when celiacs remove gluten from their diet, which allows the damaged villi and microvilli to grow back. Lactose intolerance symptoms can continue for a long time after a celiac has gone on a 100% gluten-free diet. In some cases the villi and microvilli damage can take up to two years to heal completely, but in most cases it takes between six months and a year. Most people who are lactose intolerant can usually eat goat and sheep (feta) cheeses without any problems."

Celiac.com

Truth.

A gluten-free diet does not create lactose intolerance. However, when you are doing the plain gluten-free diet and not proper Paleo (e.g. offal, coconut oil, sea veggies, bone broth etc), the gut does not get better fast-enough, so more allergies are popping up. Also, when the immune system is free from one threat (e.g. gluten), it has time to go fight other stuff too, that were always there, underlying.

I'm not a doctor, but I'd try home-made goat kefir. It must be home-made (bottled doesn't have all the right stuff in it), it must be of goat or sheep milk (cow casein is not compatible with the human gut), and it must be fermented up to 30 hours (sour, extremely little lactose remains in it). Kefir is the only fermented dairy that has a lot of enzymes to help the body break down lactose, and within a few month's time, it reverses dairy intolerance in most humans. It's also 5x-10x more potent than yogurt. Do your research about it and consider it.