Yes, you'll be able to apply for an EEA permit and it's a super simple process that doesn't require an immigration lawyer. The best way to do it is to get married and then return to the US to apply for an EEA1 permit, which is good for 6 months. It'll allow you to work, access the NHS, etc as soon as you step off the plane in the UK. If you use priority processing, it should be processed quickly. The application itself is free and priority costs $150. Then within the 6 months, you can send off for an EEA2, which is good for 5 years. That application is also free but it can take quite awhile to process. The good news is that unlike UK-track family visas, you can request your documents (passport) back while it's processing without cancelling the application and then travel outside of the country.