Unfortunately, it's usually a case by case (and locality by locality) thing. While peanuts get all the news, there are more people with allergies to other things that are just as bad as described, but who are also not given an epipen. We are in central london and were given one a few years ago - mainly because of the size of our skin prick reaction to peanut, however DS is also allergic to dairy, soy, wheat, corn, eggs, tree nuts, legumes & gourds... so probably swayed by the chances of allergies layering and causing a 'bigger' overload on the system.
WRT reactions getting worse each time, this isn't necessarily the case, so please don't be too concerned about it. Usually they stay the same for a while, plus it's also possible that they get better (DS can now tolerate baked egg without major reaction whereas he used to vomit for about 4 days if I ate baked egg, and he had my breast milk). The reaction you described is probably best treated by piriton. If it gets worse, and you do have an anaphylactic incident, I can tell you from experience that the ambulance will prioritise and be there super quick ... and they may not use an epipen anyway (eg piriton and salbutamol can work wonders!).
While we had a script for an epipen, when DS started nursery, it was a fight to get a second one to leave at the nursery. Now that he's had an ana reaction (to we still don't know what; it was probably a contaminated foodstuff), we have managed to get 2 for nursery and 2 for home (as recommended by pretty much all allergy specialists).
If you'd feel more comfortable getting an epipen, and the gp won't prescribe one, then the easiest way is to get a private appointment with an allergy specialist and get them to write a script. Be aware though, that carrying an epipen doesn't mean you won't still want/need to carry piriton. We have both at all times. Most reactions are best treated with piriton, the epipen is there in case it all goes pear shaped....
Best of luck; i do understand your concern, I'm not meaning to sound dismissive, but there is 'some' reason as to why they're not giving you an epipen....