In a few weeks, my two kids (6.5 and 4) and I will be in London. We will need to get from Paddington Station (Heathrow Express terminus) to Euston Station (trains to the North). I am trying to figure out the safest way to do this. For the purposes of this question, I am mostly defining "safe" as meaning minimizing the amount of time walking along the busy streets with two kids and all our luggage, while in a jet-lagged state. For example I don't really want to get the Circle line tube to Euston Square then cross over busy Euston Street to get to Euston Station proper.
My research so far yields the following options:
London taxi. This seems like a no-brainer, but so far I don't fully understand the carseat situation. As far as I can tell, it is legal for the kids (of this age) to travel using adult seatbelts, if carseats are not provided. However, in the interests of safety, I don't want to travel in a taxi without carseats of appropriate size for my kids. I've read that some taxis have carseats, but its not clear how common this is.
Tube. Unfortunately as far as I can tell, there are no direct tube lines from Paddington to Euston station. I think I'd have to do Circle line to St. Pancras then Northern line to Euston, or Bakerloo line to Charing Cross then Northern line to Euston. The extra tube change does not sound particularly appealing.
So my question boils down to the following parts:
Is it reasonable to expect to easily get a taxi with carseats at Paddington Station, without an unreasonable wait?
If taxis are not an option, then what's the best tube route from Paddington to Euston, bearing in mind luggage and two kids.
A bit more background: Our journey is actually from Heathrow to Stoke-on-Trent. After considering safety, I'm hoping this journey will be as relaxed and stress-free as possible, so I'm hoping for as few changes as possible. Heathrow Express seems to be the most reasonable train out of Heathrow, and Euston to Stoke on Virgin seems like the best way to get to Stoke. However if there are other options that I've missed, then I'd love to hear them.