From the Deed Poll site:
If you cannot get your [non-British] passport changed to your new name, you may find your Deed Poll will not be accepted by all UK government departments, companies and organisations on the grounds that your are not changing your name "for all purposes".
So while it's possible to change your name in the UK if you cannot get your passport to hold the same name as your Deed Poll shows it is possible that it won't be accepted everywhere. This basically means that your official name in the UK will still be the one on your Hungarian passport, and not the one on your Deed Poll until you get your British citizenship.
Note that Hungarian naming law does allow you to get a non-Hungarian sounding name in case of "exceptionary circumstances" (link in Hungarian). I don't know whether you've tried tho get these "special permissions", but if not it might be worth a try.
Once you obtain British citizenship by naturalisation you can get your name changed on the UK passport to the one that is in your Deed Poll. Having two passports from two different countries (as a dual national) with different names should not cause any problems (more exaclty it should cause the exact same issues that other people face when having multiple passports even with the exact same name: when travelling they should always use the same one when entering/leaving a country during travels, so the passport numbers in the entering and leaving logs in foreign countries match. Except of course when they travel into/from the two countries they are citizens in as in those cases they should use the respective passport).