My husband is allergic to latex and so far so good. For gross anatomy they bought their own gloves, so he just got nitrile. I think nitrile are a bit more expensive, but not prohibitively so. The hospital systems are almost all officially latex-free, but in my experience research labs rarely use nitrile and you have to request them specifically. Actually, interestingly enough, the only place my husband had a problem any time recently was with the Red Cross when he wanted to donate blood! They only had latex equipment and gloves, but thankfully it was in the school so he ran up to the gross lockers and got some of his nitrile gloves for the phlebotomists to use!
p.s. to a previous poster: I have read that there are latex-free condoms but I've never seen them. I think you probably have to special order them or go to a store specializing in..er...such things 😉