*Edit - It may be worth Staying in London for the night, sleeping in a hotel and continuing your trip tomorrow (03 July 2013) - it will be a whole less stressful and I you can walk around London for a bit
There is a train line (Heathrow Express) running from Heathrow Airport into London Paddington. The cheaper ticket is on the Tube. Buy a Zone 6 Travel Card.
*Edit - with the time that you are arriving - I would recommend the Heathrow Express as it is a lot quick getting from the Airport into London.
Once you have reached London Paddington Station, Catch the London Undergroud (Tube) Circle Line (Yellow Line) to Aldgate Station. Using the Travel Card
Walk from Aldgate Station to Fenchurch Street Station, and Catch the last train (00:31) out to Basildon (You will have to buy a single ticket out which I think is around £7.00) . There a lot of maps on street corners in London - if you are still unsure where to go, ask the underground staff
If you are very late / get lost, I would suggest getting a hotel in London for the night (premier inn, travel lodge).
PM me if you need more assistance on here.
