I land Heathrow T3 at 5:30 pm and need to rush to Euston Station to take the train which depart at 7:30pm. What would be the quickest route ? Heathrow Express to Paddington and Taxi to Euston station or .... Thanks in advance for your advice.
Heathrow express tickets are priced according to peak (7-10, 14-19 M-F) and off peak times. The off peak costs £22 for a single ticket. The tube also has peak/off-peak times/prices (6:30-9:30, 16-19 M-F).
You're cutting it very closely as you'll need time to leave to plane, (if applicable) go through customs, pick up your luggage, and then the ride to Paddington is about 15mins. I always buffer 45-60 mins if customs is included in a large international airport.
Depending on which line (Circle or Hammersmith) you take to Euston, the underground takes 15-20 mins if you know exactly where you're going.
Another local may have a more direct route but as a past traveller taking Piccadilly directly to Euston (one tube station change) is about 50-60 mins. It appears that the Piccadilly line is currently affected by delays of emergency repairs.
It should also be noted that you'll be paying more £ if you pay by cash a single way by tube instead of using an Oyster (contactless) card. That will also take a few minutes to set up and load.
It comes down to cost vs time.
Note that if you're unfamiliar, to add buffer to allow for turnarounds or asking for help.
Edit: As noted by other users: there is a bit of a walk from Euston Square to the station. Apologies, I was busy reminiscing on my time there that I forgot to note that the second link leads you to the underground webpage for travel directions.
Mdewey in comments below provided detailed directions for the walk from Euston Square to the station:
When you through the ticket barrier at Euston Square turn right which takes you under Euston road and onto the correct side of the road (which is very busy).
You'll probably hit rush hour in London, so a taxi might not be the quickest from Paddington to Euston.
Some years ago I had to get from Heathrow T5 to Euston in about two hours (I don't remember it exactly - and it was later in the day), I took Heathrow Express to Paddington, the Tube to Euston Square and walked from there, it's only a few minutes walk.