I just moved from Sydney to London about 18 months ago (and from America to Sydney about 14 years ago). This is all based on my experience and YMMV, etc.
Do you have pets that you are moving? I moved 6 (4 cats and 2 dogs). It’s expensive but not terrible, but be sure the rabies shots are up to date. PM me if you do and we can chat. You get a Pet Passport and they do not have to quarantine and they can be taken into Europe, if you want.
Furniture you are shipping will take a while to arrive. Either ship before you leave and live in a hotel for a bit, ship when you leave and live on camp chairs and paper plates, or something in between.
Bed and sheet standard sizes are different here. They were the same US - Australia. If your bed is old and so it your mattress, consider replacing them in the UK. If not, be prepared to order sheets from the US until you replace the bed, and if you have to replace the mattress think about the cost benefit of replacing the bed vs ordering one that fits.
For that matter assess your furniture - your husband’s company is probably paying to ship it but we left a lot behind because - and I cannot stress this enough - houses are very small here by American standards. Don’t be fooled when they say ‘double bedroom’ as a second or more bedroom. They have a concept of a ‘box room’ here which is a bedroom tiny enough for a small child or storage. The houses will be bigger in the countryside rather than the city, and bigger still than they are in London (the land of very tiny very expensive houses, and yet I still love it.) Get an idea of the sizes of house you are looking at (Rightmove and Zoopla are good) and then really think about what you’re taking with you. I was so very glad we did this because it saved me the stress of getting stuff ready for the movers. Like you, my husband’s company moved us, too.
Washers are in the kitchen, and you can buy a dryer but in a rental house you suffer because there’s no way to vent one. However, radiators are really ace and drying laundry. We purchased a house last year. I bought a proper American style fridge freezer. I found that living in Australia I’d gotten used to no dryer, so I didn’t bother.
Nobody has a garbage disposal, and at least one of my friends thought it was a made up thing!
Compared to Americans, Brits drink like it’s their job. Compared to Australians, though, it wasn’t a hard transition to drink less. 
You will have (potentially) trouble opening a bank account. Firstly, they are very strict on who can do so - you will need to produce visas, passports, a letter from your husband’s place of work, etc. If they are importing him his workplace can help guide you. Secondly you are American and some banks just do not want to deal with the extra reporting due to the tax treaty and FACTA. Good luck. We did ok mostly because my husband is NOT American, so he opened the account then added me onto it later.
People from Birmingham are called Brummies. I’ve been there several times and really enjoyed it. The shopping centre in the Bullring is really nice. My tattoo artist is there, if you’re into that Sort Of Thing.
The NHS is just fine. I have private insurance through work, I never use it. I have had pneumoni a since I’ve been here and the NHS emergency and hospital system was superb. I also have several ongoing health conditions, which are managed just fine on the NHS. If you are actually having an acute health condition you won’t have to wait. I’m on a waiting list to see an ENT right now due to my sinuses, but, while this is annoying and occasionally I get sinus infections, my GP is managing that while I wait. If I wanted to jump the queue with health insurance I could, but it’s not too bad now.
Plugs have on/off switches. It’s not so much an issue today but remember your American electrical things are at 110v, and the UK is 220v. Make sure you don’t blow something up.
Plumbing is…interesting (I live in South London, though, so take this with a grain of salt.) There is no bloody water pressure on the second floor of any house I have ever showered in in this country with the exception of major hotels. I have an electric shower because the house did not come with a shower. Many UK houses - particularly older ones not renovated recently - have bath only, or a bath with a shower hose sort of attached to the taps. If you like baths instead of showers, you are probably going to love this. I do not.
Speaking of taps, you often run into a sink with hot and cold taps and no mixer. No, I don’t know why.
Trains are expensive but amazing. You can go anywhere pretty regularly. Not like America at all. I love it. Also you can upgrade to first class on weekends for cheap, if you want to feel fancy.
The UK is beautiful. I think the West Midlands are lovely. It will be far cheaper to live there than in London, and London can be a day trip if you want.