What should I do if I am on a spouse visa in the UK and I want to work remotely for my US employer? What are the legal processes, if any, that I need to go through to do this? How do I report my income/employment to the UK and the US?
Normally on a spouse visa includes the right to work - so you don't need to do anything special there (sometimes there are certain restrictions - such as not working for the NHS - you need to check what your visa says).
As for working for a non-UK employer there are two options.
Firstly the US company could set up a UK subsidiary, and you become an employee of that subsidiary business. So legally you are now working for a UK employer, so you have a UK legal employment contract, your pay comes through a UK payroll, the UK company issues your P60 to report your income to HMRC. So from the UK side you just look like a normal employee working for a normal UK company. That UK company needs to be no more than a shell - all it needs is the back-office business of employing you. The US parent company periodically transfer some money to its UK subsidiary so it has funds to pay you.
Many large accountancy firms have off-the-shelf packages for dealing with this sort of thing, it is pretty common (in fact that is how I am employed at the moment). For a fee they can setup the UK company and handle all the paperwork associated with running it, so it becomes almost transparent between the US parent company and the UK based employee.
Your other option is to be self employed in the UK. So instead of being employed by the US company you are a technically a self employed contractor. You periodically invoice the US company for your services offered, and the US company pays these invoices just like they would any other. From the UK point of view this looks no different to any other self employed person. You need to maintain records of your invoices, expenses and payments, and report these to HMRC just as any plumber or gardener would.
Again any decent accountant firm can handle the book-keeping side of that if you want.
If you are a US citizen, beware the IRS respects no international borders, you will still need to fill in a US tax return every year. There is usually no problem presenting a UK P60 in place of a US W2. Things get a bit fiddly on the tax returns as the UK and US have different tax years - (I needed some professional help filling in my first couple of US tax returns after I returned to the UK). There is a double-taxation treaty between the UK and the US, so you can count UK tax paid as a deduction against your US tax bill. In almost all cases this will result in you having zero tax to pay to uncle Sam - but the IRS still gets uppity if you don't fill in the tax returns.
One thing to watch is the IRS does not recognize many UK tax-efficient savings schemes other than personal private pensions. So if you take out a UK ISA you may find the IRS will want to tax you on the interest earned.
Also you may need to take steps to keep your finances separate from your spouse. If you are not careful the IRS may decide he is a US-person, by being married to a US citizen, and decide they are entitled to tax him also.