Thanks for your question. I will try to help.
As you probably know, US cititzens are generally allowed to travel to the UK without a visa and then are granted one when they arrive.
It’s discretionary though, which means that the entry clearance officiasl can refuse where they think someone may breach the immigration rules by attempting to stay longer than 6 months. This raises issues occasionally for married persons in your wife’s position because the officials deem such persons as potentially higher immigration risks.
If you had more time before travelling then she could have applied for visa ahead of time, but there isn’t enough time now.
I would suggest that she travels with as much documentary evidence of her compelling reasons to return home. So, evidence of her job which will continue after her stay (eg employment contract, perhaps a letter from her employer stating she is required/expected to return, evidence of a tenancy agreement or home ownership, bank statements, return flight ticket, basically anything which shows that she has continuing responsibilities back home.
If you carry this evidence with you and make sure you have your stories straight that she will return home and none of you have an intention to remain permanently in the UK then it will help.
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