You can transfer to Eastern Europe. I don't recommend reaching out to the agencies, you should go directly to the Dean's office. In my experience the agencies are pretty rubbish and they charge.
Having been educated in Eastern Europe myself there are few things to bear in mind;
1. It's very difficult or impossible to get a loan for living or fees. Living expenses are much less than here though and fees range from around 7-12k/year. Often paid in Euros or the local currency and at the moment FX markets seem very reactive, meaning you cannot guarantee your fees for the next X number of years.
2. Depending on what year you are in to whether this is relevant; the UKMLA will be rolled out soon, so if you want to come back that could be difficult as I am assuming it will be based on the UK curriculum and well supported for learning here. Also, with Brexit, RN a new step has already been introduced for IMGs. They must have their diplomas verified by an outside company in order to gain a license. In Eastern Europe students graduate later and this extra step might render them ineligible if they cannot do this process before August. There is also no guarantee that this won't change again and they might introduce another hoop to jump through.
3. If you don't want to come back to the UK/US then you will have to learn the language of the country. I'm pretty sure you won't want to work in Eastern Europe, for example in Poland, "staż" the FY1 equivalent pays around 600 £ a month, it's barely enough to survive.
4. In Eastern Europe the opposite movement to the UK is happening, recently they just introduced a blanket ban on abortion. There is also an "anti-immigration sentiment". i'm also fairly conservative, but when they started closing shops (even supermarkets) on Sunday because it was in keeping with Catholic values ..... I wondered how much more I could take!
5. In terms of teaching, you get bad and good. I'm sure it's just like anywhere else. But if you perceive something as unfair nobody will listen to your voice. I didn't know one single student with reasonable adjustments in my school, it was just something they hadn't heard of. So my dyslexic friend just muddled through.
6. Due to point 1, there are very few Brits. Mostly there are Irish, Scandinavians and those from further afield. These groups are clicky and if you want to study in a group they will want to conduct it in their own language and most of the social events were geared towards Scandinavians. Personally, that meant I forged few friendships and it was quite a lonely time, although the ones I forged were of good quality and i maintain to this day !
7. At my school, exam schedules were made by students, this was carnage. Being an overseas student means there are 2 groups of students .... those desperate to get home and those who have a GF/BF at uni and don't want to go home because they live far apart from their partners.