I know that others have recommended a taxi... but I would actually disagree with that advice; if I was taking that journey, especially during the rush hour, I'd go by Tube from Euston Square.

Google's "get directions" tool claims that the taxi journey is 12 minutes. Having walked along Euston Road quite a few times my guess is that it would take longer than that because of traffic... plus you may have to queue for a taxi.

Getting from Euston to Euston Square Underground takes, at the most, five minutes (and you DON'T have to cross Euston Road as someone else seemed to think; just turn right out of Euston, walk along your side of the road and the Tube entrance is about one block away). Journey time to Paddington is 10 minutes.

If you have a through ticket from Manchester to Heathrow then this should include the Tube journey - if there is an asterisk on the ticket after the name of the destination station then it means "London Underground included".