Unfortunately, Liverpool Street station does not have lifts from it’s platforms to street level, so you wouldn’t be able to take the underground there step-free. Here are some options:
Take the bus
You can take either the 214 or the 205 between London Kings Cross and Liverpool Street Station. The 214 leaves from stop E, makes 12 stops and takes around 18 minutes. The 205 also leaves from stop E, makes 15 stops and takes around 28 minutes.
Take the Circle, Hammersmith & City or Metropolitan line and get help going up and down the stairs
It is worth noting that the Circle, Hammersmith & City and Metropolitan lines are only a couple of flights of stairs down at both Kings Cross and Liverpool Street. So if you’re travelling with a stroller and able to carry it up and down stairs (or have someone help you) then you could take this route. I always find stairs easier to cope with than escalators with a double stroller because people will tend to help you.
All three of these lines go from London Kings Cross directly to Liverpool Street, so it’s a very quick and easy journey. I should note that I’ve always had someone offer to help me carry my stroller up and down stairs when I’ve needed it. Despite the rather unsavoury reputation of Londoners I’ve found that people are generally very nice and helpful. Personally I’d take this route and bank on getting help up and down the stairs.
Go from Caledonia Road & Barnsbury to Shoreditch High Street on the Overground and walk/take the bus the rest of the way
Depending on exactly where you’re going from, this could be a viable route. If you’re literally going from Kings Cross to Liverpool Street it’s probably quicker to take the bus. However, if you’re going from or to the surrounding areas then these stations might be close enough to work well.
Caledonia Road & Barnsbury is an overground station just North of Kings Cross. They have ramps up to the station, plus lifts to and from the platforms. It’s about a 20 minute walk from Kings Cross Station.
Shoreditch High Street has step-free access from train to street. From there it’s about a 10 minute walk to London Liverpool Street Station.
Correct as of July 2017