Marriage Visa with 3 children

It is a bit disheartening, the numbers. The problem with him moving here us that he has two children as well that he would have to leave behind, which wouldn't work for him or myself. I do appreciate all of the quick responses. I'm also curious about any possible options having to do with my distant relatives there? Or possibly even a student visa as a last resort? To further something in the nursing field?

Regarding UK citizenship by descent:
 - If you're just a US citizen, only your parents can generally be "distant" enough to allow you to be British by descent. With some exceptions for recent Crown service (for example), you can't go back any further than that. The fact that your great-grandmother might have been born in Wales, for instance, doesn't help, immigration-wise.
 - If you're a US *and* somehow also a Commonwealth citizen (e.g., Canada/NZ/Australia), you can use your grandparents (your parents' parents) to apply for a UK Ancestry visa with your children as dependents if your grandparents were born in the UK islands (England, Scotland, Wales, NI, the Isle of Man, the Channel Islands and pre-1922 Ireland). But that doesn't stop you from having to pay those application fees or the health surcharges both now and over the next five years. As people have been saying, you'll still have to budget 10-12K (or more) for both yourself and each of your kids (>40k total).

Regarding other EU/EEA/Swiss citizenships:
 - What other EU et al. country/countries do you have a connection to? Some countries are more generous than others, so you'd have to be specific in detailing your ancestry. Keep in mind though: Brexit. It's a real thing with serious consequences. No one knows precisely what's in store for EU citizens in the short/mid/long-term. Personally, I'd be super cautious (and reluctant) to plan for something so important based on that when there's so much uncertainty floating about.

Based on that, can you outline your UK/EU/etc. ancestral connections and any other current citizenships, besides US?

UK Ancestry info, just in case:

https://www.gov.uk/ancestry-visa/eligibility

RE: Student visas

University courses for overseas students, which is what you would be considered, are expensive. At least 10k GBP or more for tuition alone, although you could perhaps apply for US-based student loans to cover them. You'd have limits on your term time and vacation working hours. And you also need to be careful, because some NHS-based clinical education courses require that you already be a UK/EU/EEA/Swiss citizen or hold ILR (indefinite leave to remain) at a bare minimum to be admitted to the programmes.

And on top of that, you'd still have all the visa/health surcharge fees for you and your kids to consider.

Others here can advise you on the intricacies of switching from the student visa to the spouse category in due time, if this is the path you ultimately choose.

More information about student visas here. There are important details about who's able to bring their dependents (there are limitations) and the savings involved that you need to show to support yourself and your children.

https://www.gov.uk/tier-4-general-visa
https://www.gov.uk/tier-4-general-visa/family-members

Does your fiance receive any of the following? 

- Carer's allowance
- Disability living allowance
- Severe disablement allowance
- Industrial injuries disablement benefit
- Attendance allowance
- Personal independence payment
- Armed forces independence payment or guaranteed income payment under the armed forces compensation scheme
- Constant attendance allowance, mobility supplement, or war disablement pension under the war pensions scheme
- Police injury pension

If so, for the financial requirement, you would just need to meet the adequate maintenance requirement rather than a set income.  (though the exorbitant application fees would still be charged)

Would you be selling a property if you moved to the UK?

I've been looking into positions on the NHS but I'm under the impression that I need to have started a Visa to apply to any jobs.

Are you a nurse? Nurses are on the UK's Shortage Occupation List. Visas for those who can do a job on the UK's SOL, are quite easy and very quick to get, once you have a work sponsor (NHS Trust). You can apply for these nursing jobs from outside the UK and don't need a visa to do that. Your employer would pay your visa fees. Your children would get visas as your dependants, as long as you can prove what larrabee said above. NHS England have been having a recruitement drive for the last 9 months and still are. Over this time, thousands of nurses and their families have been granted visas to the UK. Now is a good time to apply. Not all doctor occupations are on the UK's SOL.

It would be a Tier 2 General visa and that leads to settlement (Indefinite Leave to Remain) in 5 years. https://www.gov.uk/tier-2-general Exactly the same time to ILR as those arriving on a spouse visa, except that these highly  skilled immigrants and their dependants, are not required to meet the minimum financial requirement for settlement (ILR) in the UK. 

On a Tier 2 General, the NHS will be your sponsor and not your husband: which means that if your marriage didn't work out, it wouldn't affect the  visas of you and your children.

Information on registering as a nurse in the UK:

https://www.nmc.org.uk/registration/joining-the-register/trained-outside-the-eueea/