Be sure to take your epi-pen, you won't have issues with it at customs as they are familiar with them. In Havana, the tourist hospital is:
CLINICA CENTRAL CIRA GARCIA
Calle 20 #4101 Esquina Avenida 41
Playa Havana Cuba
TEL ++53 7 204.2811
FAX ++53 7 204.2640
email : faculta(at)cirag.cu
http://www.cirag.cu (Cira Garcia Havana)
Here is another poster on the same subject
Another helpful link:
Other suggestion, although Cuba rarely check to see if you have travel health insurance, it would be a wise option in this scenario. Sometimes your travel agency already includes it in the ticket (confirm this), but you can also get it at the airport upon entry. Make sure you as before you leave the customs area, this can run about $3.50 per day/person. If you purchase it in person, they will make sure that EVERYONE in your group has to get it. You might be able to get it online at www.asistur.cu here is the pricing http://www.asistur.cu/SEGOB_ESP.pdf
Now, I don't know if you have seen this story, but this is a friend of my daughter. He had one of the most severe peanut allergies known, and he is now eating peanut butter sandwiches and more without any reactions. ABC did a story on him and you can see it here: abcactionnews.com/dpp/news/health/Fighting-p…
While in Cuba, if you take a local taxi called a Collectivo, or jump on a bus, you may be exposed to peanuts since the you aren't the only person and there are peanut vendors.
I know your worries, and I'm sure you will have an awesome trip.