East Midlands here (Swadlincote)
We have a Sainsbury's, Morrisons and a Tesco all within 10 minutes of us by car. We even have a Sainsbury's Local that's just a short (3 minute) walk away. To be honest, we haven't gone to Tesco yet. I'd like to, but we keep forgetting!
Our typical weekly shop gets done at the big Sainsbury's, my husband and I prefer their meat selection to Morrisons. The Sainsbury's vegetables and fruit are pretty good as well. They're pastries and cream cakes are the best of the two stores. As to American products... I've honestly given up on getting a lot of them. I can get my Heinz ketchup, Pepsi and Walkers (UK name for Lays) brands at either store. They also have Knorr products, I use the stock pots for chicken or beef stock.
We go to Morrisons on occasion because I love the ice lollies they have, you get 5 for a quid. And they have chicken crowns (just the breasts on the ribcage) as 2 for 5.
Sainsbury's will match Tesco and Morrisons prices for Brand name items, so if they have a better deal on what you buy, you get that price automatically. They also have where you get money on a voucher for the next time you go, but I can't remember why and I don't have any right now. haha.
We went to our local Aldi and I didn't like it. There wasn't a lot there that I would typically buy.
And FYI, Asda is owned by Walmart. Husband says it's like it always was before they were bought out, but if you have a thing against Walmart like some people I know, I'd steer clear.
There are A LOT of websites here in the UK where you can get American products, they're just expensive and you have to pay shipping. Sometimes they offer free shipping with a certain purchase amount. I hardcore looked into that when i first moved here. There isn't a lot that I miss.