Is it unreasonable to ship my vehicle to the UK? It's my understanding the local vehicles are substantially smaller than the US (and gas is way more expensive). Honestly, seeing that my minivan is 12 years old now, I don't know if it would be worth the cost to send it over, but at the same time...if we chose to live in the London 'burbs, would we have decent access to public transport into the city? The subway system is only inside the city, correct?
I personally wouldn't bother to ship a car because of all the costs... not just shipping, but all the adjustments that have to be made to make the car 'road worthy' to be driven in the UK. The process (with shipping) is likely to cost you a couple of thousand dollars... and you could probably buy a decent UK car for a similar amount (you can get used cars here for as little as a few hundred pounds... most of the used cars I saw when I lived in the US were about $4,000 upwards and were much older than the cheap used cars I've bought in the UK!).
In terms of public transport, the underground services go right to the outskirts of London (the suburbs) -
this map shows how far it reaches... zones 1 and 2 are basically central London and zones 3-6 are the suburbs, and there are lots of overground trains in and out of the city, as well as buses too - I can get into London easily on either the train or the bus (I think they go every 30 minutes) and I live on the other side of the country

. If I lived in London I probably wouldn't bother to get a car - you have to pay a charge to drive into the city and the public transport is so good that it probably wouldn't be worth the hassle of having the car (i.e. I would probably only use it if I was driving to another city elsewhere in the country and it would just sit on my driveway for the majority of the time). As you're a family with kids though, a car might be useful, even if you're in London

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But given that my minivan has a 26 gallon gas tank and gas is what....roughly $7 a gallon over there....yikes. And I'm cringing as our prices here come dangerously close to $3/gallon.
I heard on the news last night that prices are expected to go up to £1.36 per litre... that's £6.19 ($9.87) per UK gallon, or £5.15 ($8.22) per US gallon (UK gallons are 4.55 litres, US gallons are 3.79 litres)!
What about those of us who can't drive stick shifts?
You can buy automatic cars in the UK, but they are less common than in the US. You will only be able to drive on your US licence for the first 12 months that you live in the UK... and you will have to pass the UK driving tests to drive beyond those 12 months - if you take the test in an automatic, you will only be legally allowed to drive automatic cars, but if you take it in a manual (stick shift), you can drive manual or automatic cars.