Just to add to what SimCityAT has provided:
The UK is currently in Covid lockdown, so until that finishes, you wouldn't be allowed to enter unless it is for one of the approved professions; this link will take you to the list of Jobs that qualify for travel exemptions on the UK Government website.
They are expected to make an announcement this afternoon on the plan to release the UK from Covid lockdown; it's a 4 phase plan that will likely run into the spring; the one thing that is concerning many is it does not mention anywhere when they will permit international travel. In your position, I would not be making any travel plans for the next couple of months at least and it will almost certainly depend on how well the US is doing. On a related note, there is increasing talk of Covid vaccination passports; the UK Government have admitted they are discussing it with other countries, you may well need to have been vaccinated and issued with a Covid vaccination passport before they will permit you to travel/enter the UK; the fact that international travel does not feature in the UK Governments lockdown plan is giving this increasing credence.
Pets are an issue; apart from the hassle of finding a landlord that will allow them in your rented property (many don't), the UK does not recognise the US thing of companion pets, so there are no incentives or requirements to allow any animals to travel in the cabin of any UK registered aircraft or any flight entering the UK. Your suggestion could work; your pets will need an EU health certificates to enter the EU. Brexit has certainly complicated the travel between any EU country and the UK. The UK is no longer part of the EU pet passport scheme. My advice would be to speak to the companies that deal with it. I saw a thing on YouTube where a couple travelled back to the UK from the European mainland, overland with their dog and it was a nightmare, but they made it, so it's doable.
The UK COVID vaccination scheme is going well; they are forecasting that the entire country will be vaccinated by August 2021. Were you resident in the UK, so part of the UK's NHS system for healthcare, then vaccinations in the UK are allocated on a need basis, so if you're in one of the at-risk categories, you'll get the jab, if you're not, you'll wait till all those in a risk category above yours has had theirs; this link will take you to the UK Gov website that lists these risk groups; from what you've told us, you would be in Risk Group 10. High street pharmacies are also delivering the vaccination, but only as part of the NHS scheme, so you can't walk in and buy one.
With regards to costs; the NUMBEO website offers regional prices for specific key items; this link will take you straight to there website; I've set it up to do a comparison between Columbus OH and London. That salary you mention is not exceptional, neither is the relocation package they are offering.
Hope this helps.
Cynic
Expat Team