Work Permit or EU Family member Residence permit

Jellea wrote:

I'm certain that I can convince her to live in another European country.

Why is she interested in the UK? Because of speaking English? Then why not Ireland as another English speaking option?

I wouldn't rule this option out as one of the reasons.
Additionally, she has relation living there and has been there twice. I will check out Ireland's possibility. Thanks

I read your post but not sure what to say.
First off not to get personal but why would your fiance leave without you to the UK?
Why is she in a big hurry to go?
Can she support you while you are getting your work papers in order?
My US sister was married to a UK citizen and living in London, she had to legally wait one year before she could work there after their marriage.This was in the 1970's though . All info keeps changing so really only immigration has the answers for you.
I  know from my experience as a US citizen married to a US/HU citizen that for the first 5 years of my holding a resident permit for Hungary that I was only legally allowed to leave Hungary without telling them I was on holiday for up to 90 days.
Now holding a 10 year permit and can stay out of Hungary for up to one year without going through immigration.
At least that was in the packet with my cards.
So if that is the case and you plan on moving to the UK, then you would probably have to go through their immigration services since you are not married yet to a HU citizen so that card will not work at HU immigration yet.
You would have to apply for a work permit in Hungary at immigration and hope they give it to you.
I do not see however how a work permit for Hungary will help you as a non EU citizen in the UK.
They tend to do things slightly different in person at immigration as opposed to what their web site may state.
You hold a student visa and that's different then a work visa in Hungary. You may have to do allot of new paperwork and be required to stay in Hungary and work there instead of getting a HU work permit and running off to the UK to try to work, they may not take that work permit from HU since you are not a EU or HU citizen. It can get complicated.
As it is, you are not yet married to your fiance and you have not lived together long enough to be considered a "couple" in Hungary, think it is 2 years required as a live in couple. Guess they would ask for signed documents to prove you actually lived together 2 years or longer.
The best move is to go into immigration and ask some questions, every case is handled differently depending on each situation.
Best of luck.
Immigration is a hassle in every country.
ATM my HU/US son is married to a Japanese national , been over 5 years he has paid allot of money and filed all the right papers for her US citizenship, thought she would of been called in already to take her tests.
Been waiting and waiting and running out of time. They now plan on moving to Japan where my son can legally work but seems they will forfeit the fees paid already to US immigration.
Slow going here in the US. They are tired of the wait.
To be able to even apply for her US citizenship they had to bring in allot of personal photos of them to show they really were together so long even with their marriage documents , airline ticket stubs to prove they meet as they said they did and signed legal papers from people in the US who know they are a couple.
It is almost racist the way they did this to them. Even all their legal papers were not enough to show they are a actual couple. Makes me wonder what sort of immigration scams have been going on here in the US to make things so difficult for even legit people.

Your family experience is really strange and dishertnning. 
My fiancee don't really know much about no eu plights, since she can easily go anywhere with no papers and processes.
I think the issue of 2 years couple is only for non married patners.

Jellea wrote:

Your family experience is really strange and dishertnning. 
My fiancee don't really know much about no eu plights, since she can easily go anywhere with no papers and processes.
I think the issue of 2 years couple is only for non married patners.

I'm pretty sure couples have to prove they have been in a actual relationship for at least 2 years for Hungarian immigration services.
I read it online with their info site awhile ago.
Even for legally married couples if one person is not a EU ciziten then they have different rules, have to prove you will not need any gov. aid by showing a certain amount of funds in a local bank and have to show income.
It's not easy.
Don't feel too bad it will work out at least you are physically in Hungary so it is more complicated to get you out.
It also seems to depend on who and how they handle your immigration case.
Some workers are a bit more flexible then others.
My grandfather was born in the USA in Penn. as the son of Ruysn immigrants. He went to Poland as a teenager to visit family and was drafted into the war, the Russian revolution.
War broke out while he was visiting Poland.
Got married in Poland and lived for several years in communist Poland where my father was born.
He decided to return to the USA and later brought over his wife and 2 children.
Had to borrow money from relations in the US for all of their passage to the US.
When he arrived alone first in NY they almost didn't take him back into the USA!
My first cousin just told me this story a few months back.
The US immigration said to him, "You've been gone too long"!
He got in but it just shows how lamo some people with official jobs can be, what a maniac to not think my grandfather should not be let back in! It's a good thing to have rules because sometimes they do work for you and not against you. If it was up to that silly immigration man, I'd be a Polish citizen right not and not an American. If I was born at all. It will work out but first you may have to get married in HU or talk your lady into slowing down with moving for a bit.

I will be hopeful. I'm really stressed out about this

Don't allow yourself to stress too much. Life is full of stressful situations. It really does all work out in the end.
Best of luck .