London, Euston and London, Victoria both have underground stations on the Victoria Line.The journey time is about 8 minutes.
About 20
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.
50 minutes
About 25 minutes by cab.
Birmingham International - Euston, then take the Northern Line to Waterloo, and the Jubilee Line to North Greenwich for the Dome.
About 2 hours - but it depends on traffic conditions at the time.
The best option is to walk to Euston Square Station - which is well signposted and a two minute walk from Euston Station. Then, take the Hammersmith and City Line, Circle Line or Metropolitan Line to Liverpool Street.
The most direct path is east to the A201 south, across the Blackfriars Bridge to Southwark and then "The Cut" west to Waterloo Road. It is 6 minutes faster, on average, to use the A4200 due south across the Waterloo Bridge.
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
The Trainline web-site says - take a tube train from Euston to Paddington, then a first Great Western service to Reading. Journey time Euston to Paddington is 28 minutes and Paddington to Reading is 25 minutes.
The quickest way is to take a train from Euston Station.