Depending on the time of day and congestion, the taxi journey between Liverpool Street and Euston stations in London should take 20-30 minutes.
A more reliable route would be the Underground, using either the Hammersmith & City, Metropolitan or Circle lines one stop to Moorgate than three stops on the Northern Line.
I would allow 20 minutes
London, Euston and London, Victoria both have underground stations on the Victoria Line.The journey time is about 8 minutes.
About 20
Like most journeys in London, the best way is to go by tube (underground) or you could take a cab.
The quickest way is to take a train from Euston Station.
No, Paddington - to get to Euston, you need to take the Bakerloo Underground line from Paddington to Oxford Circus, and then take the Victoria line to Euston
Take a South West Train service to Vauhall and then change onto the London Underground Victoria Line which will take you to Euston.
About 25 minutes by cab.
Both stations are on the Victoria Line of the Underground railway system so no need to change, just take the Victoria Line southbound.from Euston. If you have to drive then from Euston Station take ... Euston Road, Gower Street High Holborn Shaftsbury Avenue Charing Cross Road Trafalgar Square The Mall ( Through St James Park and past Buckingham Palace) Buckingham Gate Buckingham Palace Road Victoria Station. That's 2.9 miles
About 15 miutes by cab or 45 by public transport.
Take the Victoria Line direct from Euston to Victoria. Alternatively, take a cab - however, it will be considerably more expensive and probably won't be quicker.
The distance between Euston Station and Waterloo Station in London is approximately 2.3 miles if you follow the A4200 which would take about 10 mins (excluding traffic). Alternately, you can follow the A4200 and then switch to the A40 which is approximately 3 miles and would take around 13 minutes without traffic.
There is no mainline station at Heathrow. You would have to take the underground into London Euston and get a train from there. You will have to change trains at least once and the journey from Euston should take between 5 and 6 hours.