The London subway (the Tube) runs through the airport and is the least expensive way to get to downtown. It will take a bit longer than the Express, but it is worth it. There are maps and signs will help you when you land. The Tube is the best way to get anywhere in the London area.
Pick up a Tube map and find "Tower Hill" station. You will have to change trains but it is easy.
Google for a Tube map. Also Google a map of the Tower Hill area. You can do that right now.
I think previous reviewers have probably answered your question. It might be a good idea to look at the website and contact Andrew, the manager, either by email or phone call with any queries. He's extremely helpful and doesn't mind a bit.