Hi, welcome to the forum

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If she is just coming to the UK to marry you, but will return to the US after the wedding to finish school, then she will need a Marriage Visitor Visa to get married.
The Marriage Visitor Visa costs £87 and is just a regular 6-month visitor visa, but one that allows her to marry while she is in the UK. She must be able to show that she is only visiting and that she will return to the US afterwards - so, for example, she can show she is enrolled in university and will be going back to finish.
See:
https://www.gov.uk/marriage-visa/overviewThe fiance visa is for people who are moving to the UK to marry and then stay permanently without going back to the US. For the fiance visa, she would come to the UK, get married and then switch to the spousal visa equivalent from inside the UK (called FLR(M)). If she needs to go back to the US to finish school, it sounds like the fiance visa isn't the right option for her.
See:
https://www.gov.uk/join-family-in-uk/overview (covers both fiance and spousal visas)
Just a note on terminology: the 'marriage visa' IS the 'fiance visa' - the official name for a fiance visa is a Marriage Visa (a visa for the purpose of getting married)... so for a fiance visa, you would select 'Marriage' as the visa type when applying. If you are already married, it's called a spousal visa and you select 'Wife' or 'Husband' as the visa type.
So the possible visa options are:
- Marry in the UK on a Marriage Visitor Visa (minimum 35-day waiting period: 7 days residency, then 28 days after giving notice) then return to the US, finish her studies and then eventually apply for a Spousal Visa when she is ready to move.
Visa cost = £87 (marriage visitor visa), plus £1,795 for the spousal visa (£1,195 plus £600 NHS surcharge).
Total visa cost (current prices) = £2,062
- Move to the UK permanently on a Fiance (Marriage) visa, get married in the UK, switch to FLR(M) spousal visa, which will allow work and study in the UK
Visa cost = £1,195 (fiance visa), plus £1,311 for the FLR(M) visa (£811 plus £500 NHS surcharge).
Total visa cost (current prices) = £2,506
- Marry in the US (i.e. you fly there to marry her. You won't need a visa to do this and there's no long waiting period). She then applies for a Spousal Visa when she is ready, and then moves to the UK.
Total visa cost = £1,795, plus flights to and from the US for you