Hi, welcome to UKY!
I'm pretty? sure we meet them now, but I don't know anything.
As your husband has been living with you in the US, if you wanted to rely on income, he would need to show:
- he made the income requirement in the US for the past year; AND
- he has a job confirmed in the UK to start within 3 months of moving over that also meets the income requirement
Or, if you have £62,500 savings held for 6 months he wouldn't need confirmed UK employment. Are you selling a property? If so, you can use £62,500 from that sale, and it doesn't have to be held in a bank account for 6 months.
Otherwise, he may have to return to the UK before you to secure a job so you could apply for the visa. This would unfortunately mean though that you would have a lengthy separation, which probably isn't ideal.
...and in three weeks we will have another!
Congratulations!
I don't even know what visa to apply for. I would assume the spousal visa ILR ... is that even a thing.
ILR - indefinite leave to remain, essentially permanent residency. You can apply for this after being resident in the UK with your spouse for 5 consecutive years.
The path is:
Spouse visa for 2.5 years.
Further leave to remain for 2.5 years.
Infinite leave to remain
UK citizenship
Overall, we recommend to budget at least £10-12k for the multiple visa fees over 5 years. It's ridiculously expensive.
Also, I know our children are citizens by descent but idk how to prove that. Do they just need a UK passport? They will both have US passports, so is it really as simple as sending off the US passports with the UK passport application? I believe I would be the only one to need a visa.
Yes, assuming your husband is British otherwise than by descent, your children together are British by descent.
In this case, you would be the only one to need a visa, and the income requirement would be one applicant and no dependents.
Another thing, how long does the process usually take?
Once you submit the application, I would allow a a few months for it to be processed if you purchase optional priority processing, which puts your application at the front of the queue. Without priority processing, plan on double the time. Summer is usually the slowest time for processing, so if you can, I would avoid that period to submit an application.
If we meet the income requirements, are there a lot of other hurdles that we will need to address living outside of the UK?
Credit will likely be problematic if he's been away for a while. And as a newly arrived expat, you will have to build up your credit score from scratch. Getting a job might be difficult, as previous UK job experience is important for employers nowadays. If he doesn't have established previous UK work experience, he might find it changing to find a UK job, even as a UK citizen. I can't speak personally on this, but look into potential differences in schooling; being able to transition your child from the US to UK system.
More than likely we would live with his parents for a month or so while we get settled. We have played around with the idea of my husband going ahead to secure a job and housing, but I don't know if that would make a difference.
Living with parents is absolutely fine. There isn't a requirement to have your own place.
He may need to go ahead to secure a job to meet the income requirement, but that depends on if you can meet the requirement another way (e.g. savings).
The current US administration is volatile towards immigrants
Unfortunately, I don't think you'll find the UK is any better.

The UK is quite anti-immigration, and has been for some while.
**edited to add that I have a master's degree from Columbia University in Clinical Psychology. Does this help my chances at all of securing a visa? **
In regards to a spouse visa, no. With few exceptions, like holding significant savings, only the UK sponsor's finances can be considered. The applicant's potential earning can't be taken into account.
If you were looking to use your skills and career, you would need to look into a work visa for yourself.
It is a lot of info, but breathe and take your time.

ETA - totally beaten by speedier typists!