Do you take any sort of allergy medication already, since you mentioned having allergic reactions to your cats? I have pretty bad seasonal allergies (just to plants/pollen, no animals) and I take Claritin-D (the kind you have to get behind the pharmacy counter) and use a Rx nasal spray (Flonase) every day. This helps a lot with my day-to-day symptoms, although I don't have any trouble with hives or any other skin issues.
And now, for some medical-nerd speak about allergies and allergy treatments.
The kind of treatment you're talking about (allergy shots) requires a test to see if you actually are allergic to the specific allergen (i.e. the hedgehog dander or saliva) first. These types of tests, usually done by pricking your skin or testing your blood, measure your immune system's reaction to various allergens via IgE levels. IgE (immunoglobulin E) are antibodies produced by your immune system that are specific to the allergen. The higher your IgE level for say, cat dander, the more severe your allergic response - sneezing, runny nose, hives, etc. What is actually happening is that the high amounts of IgE in your system are stimulating other immune system cells to release histamines, which, while helping to rid the body of the allergen, also cause most allergy symptoms. This is why most allergy medicines are marketed as anti-histamines.
Point being: I highly doubt most doctors have a stock of hedgehog-specific IgE on hand, so it will be hard to test if you are conclusively allergic to hedgehogs or not. Also, allergy shots are very time consuming (you usually have to go for several visits in the beginning, and every few months afterward) and expensive. But, they are probably the most effective treatment for severe allergies, since they attack the route cause, and not just the symptoms.
If I were you, I would meet with an allergist and see what they think. There are lots of prescription options out there, which would be a good first step, if you aren't taking anything already. And hygiene is super important - make sure you wash your hands (and arms) a lot to keep any excess fur or skin off of you when you're not playing with Milly. I'm sure there are also topical creams you can buy to help with any swelling/itching on your skin.
I hope this helps! My apologies if you're already fluent in the science-y stuff.