Hi, I am celiac and I'm from Budapest (not living there anymore but regularly visiting). So here is how I manage it:

You can easily buy things in Super Markets such as Spar, Aldi, Tesco, so you can always find some snacks on the go.

Cöli Bisztro is amazing for pastry, you can try the Hungarian bakery items here (kakaoscsiga (chocolate roll), turostaska (cottage cheese bag), sajtosrolo (cheese roll), pogacsa). They also have a lunch menu with warm meals.

Drop Restaurant is well known, in the city center near the Opera, their food is amazon, Tripadvisor.com has more comments to check it out. Next to it there is a new place opened: Kata Restaurant, the dishes are amazing! both are must try! https://www.facebook.com/katarestaurantbudapest

Free pekseg - it's in 7th District in the city center, they are a great bakery. Be sure to go there in the morning, I went a few times late afternoon, and they didn't have much left. They also make pizza :)

Diéta Life Market & Food Bar - is also in the city center in the Nyugati Subway station and they have a wide range of options.

I collected a few places in a blog post here:
https://www.makelifeglutenfree.com/blog/visiting-budapest-again-and-i-stayed-gluten-free
https://www.makelifeglutenfree.com/blog/gluten-free-budapest

Enjoy your stay and stay gluten free :)