Welcome to the forum and good luck with your plans
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I think your main deciding factors for which path to take will need to revolve around your visa options, cost and timescales.
There is no visa for either country that just allows you to live together for a bit to try it out and see which country you like best (there's no 'let's shack up' visa, as we call it here on the forum
).
In order to qualify for a visa to move to either the US or the UK based on your relationship, you're going to need to get married - either before you apply for any visas, or there are 'fiance' visas that allow you to marry within a certain number of days or months and then switch to a spousal visa after the wedding.
Moving to the UK - Karen and her daughter will both need settlement (family) visas to move to the UK
- in order to qualify for the visas, you must meet a minimum income requirement of £22,400 per year, which you must have been earning for at least 6 months before the visa applications are filed
- the visa applications filed in the US can take anywhere from a few days (if you pay $565 per application for priority processing) to a couple of months to be processed
There are two possible visa options:
Fiance visa- valid for 6 months
- costs £956 per visa, so £1,912 for both Karen and her daughter
- you must marry within those 6 months (and show plans to marry in the visa application, such as a provisional ceremony booking)
- no work or study is allowed on the visa
- no NHS access is allowed, so they will both need private insurance
- once you are married in the UK, Karen and her daughter will need to switch to FLR(M) visas to stay in the UK
- the FLR(M) visa is valid for 30 months
- each FLR(M) visa costs £649 (though this will increase in April) plus £500 NHS levy
- the FLR(M) visa can take anywhere up to a few months to be processed by post, or you can pay an extra £400 per application for in-person, same-day processing
- the FLR(M) visa allows work, study and NHS access
Total visa cost for fiance visas and FLR(M) visas: £4,210, not including faster priority processing (currently $565 each) or in-person processing of FLR(M) (£400 each)
Spousal visa- you marry in the US while you are there (or another country)
- once you are married, Karen and her daughter can apply for spousal/child dependant visas to move to the UK
- the visas are valid for 33 months and cost £956 + £600 NHS levy per person
- the visas allow work, study and NHS access
Total visa cost for spousal/child dependant visas: £3,112, not including priority processing ($565 each)
See here for information on the visas and how to apply:
https://www.gov.uk/join-family-in-uk/overview_________________________________________________
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Moving to the US- the US visa process is a bit more straightforward and the visa is cheaper
- however, it's a much longer process and it takes at least 6 months, with an average of 8-10 months, for the visa to be processed (compared to a few days/few weeks for UK visas)
- as with the UK visas, there is the fiance or spousal visa option
US Fiance visa (K1 visa)- allows you to come to the US within 6 months of the visa being issued
- you must marry within 90 days of entering the US on the fiance visa
- the visa costs $265
- to apply for the visa, Karen will need to be earning at least 125% of the poverty level
- she will first need to file a petition to sponsor a visa in the US (takes about 4 months to be approved), then you will need to attend interviews and medical exams at the US embassy in London (another few months in total)
- once you are in the US and married, you then apply to adjust your status to a spouse of a US citizen, which will be conditional for the first 2 years
US Spousal visa (K3)- Same process and I believe it's the same cost as a fiance visa, though there may also be USCIS fees
- If you have been married for less than 2 years, you will be given a conditional resident visa (CR) status and will have to apply to remove the conditional status 90 days before the 2-year anniversary of you entering the US on the visa
See here for information on the visas and how to apply:
http://london.usembassy.gov/mobile/immigrant-visas/marriage-to-a-u.s.-citizen.html_________________________________________________
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So, those are the basics of the visa options based on your relationship
Essentially, the UK visas are quicker to be processed (some people recently got their visas in less than 2 weeks using priority processing), but more expensive and have more requirements, while the US visas are easier to qualify for and much less expensive, but take several months to be processed and issued (so you should start the application process about 10 months before you wish to move).
Other visa options, if the relationship/marriage visas are not suitable, would be a student visa to study in the US or the UK, or a company-sponsored work visa... but student visas are expensive (you need full tuition and living costs available) and work visas can be extremely difficult to qualify for.