I could make do on £200 a week for the three of us, no problem.
We spend about £80 a week on food and household items and that's without "watching" what we spend. We could reduce that if we tried.
We only eat out about once a month. That's been a big culture shift for me in the UK. My husband thinks we eat out a lot. I think we NEVER eat out! Eating out is definitely more expensive here (we had our second date since our daughter was born Saturday. We each had a starter and main. He had a beer and I had a glass of wine. The bill was £70.) With a kid, we normally eat out at lunch so MUCH less expensive as definitely no frills restaurants with kids!
I'd recommend looking at an annual National Trust membership. It costs about £90 but gives great days out for the family, plus events at Christmas, Easter, summertime, etc. Softplay places are great for rainy days and the three of you can go at peak time for under £10.
Is £200 a week after setting money aside for return trips to the US and travel for holidays? Depends on your family's desire for return trips home and travel. For us, that's a priority.
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I think you'll find that after 6 months, you'll be fully adjusted to your new lifestyle. You'll probably wonder what you did with all your disposable income in the US!