Paddington to Euston with kids: minimum walking by tube or by taxi?

If your purpose is to get from Heathrow Airport to Stoke-on-Trent with as little hassle as possible, I would consider flying to Manchester and then take the direct bus from Manchester Airport to Stoke-on-Trent.

British Airways operates about 8 flights a day from Heathrow to Manchester and there are seven daily buses from Manchester Airport to Stoke-on-Trent operated by 'National Express', most taking only 50-55 minutes.

Having traveled from Heathrow towards London by tube many times, my first reaction was: Why to travel by train as well as tube.
Starting at the airport you will be one of the first in the tube and you will have an easy time to find seats.

Look on the transport for London site for the best travel options on the day and time you need to travel, set it to 'least walking' to get it to bring you to the Euston underground station if needed.
I see an option to take the Picadilly line to Green Park Underground Station change there for the Victoria line till you reach Euston station. Should take about an hour.
Do check for stairs, if you are walking with a lot of luggage you do not want a change where you have to climb a lot of stairs.

If you also want to do some sight seeing, you could go for a bus part of the way, but it is likely going to be more expensive.

When planning to use a taxi, book a taxi that will consider your flight arrival time, they might be a little more expensive but they will be ready for you when you arrive, they should be able to have child seats as well.

There is a direct number 205 bus from Paddington to Euston. Ordinarily it goes down the side of Paddington (platform 1 side) and then along the front of the station. It calls into the bus station that's adjacent to the concourse at Euston. It takes 22 minutes for the journey, and buses are every 6-10 mins during the day, less frequent early and late. The bus is door-to-door, and it shouldn't be an issue to have luggage (though, like all London transport, it'll be busy at peak times). The 'iBus' system calls out stops so you'll know when to get off - ring the bell to indicate you want to get off at the next stop.

TfL buses are cashless, so you'll need either Oyster or contactless bank cards to buy tickets. (You can also buy a paper travelcard, but it's not worth it for the £1.50 bus fare)

I say 'ordinarily' because it's diverted a few streets away from Paddington until 12 July due to roadworks so will only pass by the side not the front of Paddington until then.

As you are aiming to get to Stoke-on-Trent, you can take a bus from Heathrow to Watford Junction, then take the train. The train may be cheaper and you avoid the center of London.

Or take the Tube directly from Heathrow to Euston, as you are getting on at the end of the line you should get a seat.

I would be inclined to take the tube to Euston Square - it hasn't been mentioned yet, but it is possible to exit Euston Square station on the north side and take the short walk to Euston station without crossing any roads (indeed, you can use the station as an underpass if you need to cross the road).

There are a couple of short flights of stairs involved (the lift to street level is on the opposite side of the street so would involve crossing the road), but should be fairly manageable.

One thing to point out about crossing the road which you may not appreciate as an American is that pedestrian crossings in the UK are a bit better controlled than they are in the USA - there are no right/left turns permitted at red lights, if you have the 'green man' indicating it is safe to cross then the traffic is all held - this is the situation at Euston Square.

If you are leaning towards taking the taxi, I would reassure you that it is a very short journey and the statistical likelihood of a car seat being required or making a difference is tiny. One thing to point out is that two of the seats in a standard black cab are rear facing, which makes them safer in the event of an accident - if you put your two smaller children on these then you are making the trip that bit safer for them - just make sure you are all wearing seat belts.

If that doesn't appeal, then as mentioned you should be able to pre-book services you require.

(If cost is not an issue) You can get a pre-booked taxi/private hire to take you from Heathrow, straight to Stoke-on-Trent. As mentioned in comments your best bet is to book a firm/individual who resides in Stoke as this will be cheaper (you may need to pay a deposit or even the whole amount up front)

Do not just jump in a black cab outside Heathrow and ask them to take you to Stoke as this will be eye wateringly expensive

It will take around 2.5 hrs to complete the journey. Any other travel option will probably be at least double that time door to door.

The cost will be around £180 (one way) but this will be by far the quickest and least hassle option.

in comparison; flights from Heathrow to Manchester will cost upwards of £80 for your family and train fares will be £50 minimum from Euston to Stoke. Add on your travel costs to get from Manchester airport to destination or to Euston from Heathrow and it really isn't that much more expensive to go by taxi.

You could also hire a car at Heathrow and drop off in Stoke (there are plenty of big name hire car outlets in Stoke). This will be cheaper (somewhere in the region of £75+ for a days hire) but as you specifically mentioned being jet lagged you may not want to go down this route.

If convenience of journey is more important than time taken, you can get from Paddington to Stoke by train with one or two changes. One at Reading, and possibly one at Birmingham. There should also be step-free access between platforms at both stations.

Depending on timetable, it could take as little as half an hour longer, or possibly up to about 1.5 hours longer.

If you take the Railair bus from Heathrow to Reading, instead of going to Paddington, you can save yourself the trip into London (and associated expense). This might add an extra 10 minutes onto the "perfect" time, (Heathrow->Reading: 50 minutes, Heathrow->Paddington: 15 minutes, Paddington->Reading: 25 minutes), but that doesn't factor in time taken to change or wait for connections at Paddington, so could be even quicker, depending on the timetable.

The train at Reading will be a Cross Country train and will not have come from London, so should be less packed than the Watford option.

It could also mean you do the whole journey with as few as two simple changes instead of at least three (one of which might be difficult). (LHR-RDG-SOT, vs. LHR-PAD-EUS-SOT)

Can you fly to Birmingham instead of London? There is a direct train service (hourly) from Birmingham International to Stoke-on-Trent, 75 minute journey. That train will be considerably cheaper than Heathrow express plus a train from Euston.

Just make sure you don't make the mistake that wasd recently reported and confuse Birmingham Alabama with Birmingham England!

I did a similar transfer last fall, with a 4 year old and 2.5 year old. It's entirely possible, if you are careful about how you navigate the station to avoid steps. Highly recommend the step free tube guide for that - we got caught once with steps when we didn't, and we won't do that again!

Black cabs are perfectly usable with kids. They don't necessarily have proper car seats, but the seats are quite usable even with smaller children - we didn't have carseats but the middle seat worked for our youngest, and our oldest (about the size of a normal 6 year old, he's big) fit fine in a normal seat. There are rear-facing seats in the black cabs; that can be helpful since rear-facing is safer in accidents. With kids at the age you have, I think you'll find the safety issues are minimal - and if you're really worried, bring boosters (Both are old enough for just boosters) which will be free to check on the airplane.