Travelling to Central London
From London Heathrow (LHR)

Heathrow Airport is located 32 km (20 miles) to the west of Central London.  The primary methods to get into Central London from Heathrow are:

Train

Heathrow Express to London Paddington - around 15 minutes
Tickets £16.50 to £23.00 single - http://www.heathrowexpress.com

The Heathrow Express is the fastest way to travel into Central London.  Trains leave Heathrow Airport between approximately 05.00 until 23.45.  Standard adult fares are £16.50 for a single ticket purchased online (£18 if bought from the ticket machine at Heathrow Airport and £23 if purchased onboard).

Heathrow Connect to London Paddington - around 25 minutes
Tickets £8.50 single, £16.50 return - http://www.heathrowconnect.com

Heathrow Connect services leave from Heathrow Central (Terminals 1 and 3) and Terminal 4 stations and call at Hayes & Harlington, Southall, Hanwell, West Ealing, Ealing Broadway and then Paddington. 

London Underground / Tube

London Underground to Central London - around 50 minutes (to Piccadilly Circus)
Tickets £5.00 (£4.50 Oyster) - http://www.tfl.gov.uk

The Piccadilly line connects Heathrow Airport to Central London and the rest of the Tube system.  The Tube is cheaper than the Heathrow Express or Heathrow Connect but it takes a lot longer and is less comfortable.  Tube services leave Heathrow every few minutes between approximately 05:00 (05:50 Sundays) to 23:45 (22:50 Sundays).  There are three Tube stations at Heathrow Airport, serving Terminals 1-3, Terminal 4 and Terminal 5.   If you're going to be using London's public transport system, it's worth buying an Oyster card.

Taxi

Licensed Taxi to Central London - between 30 minutes and 1 hour
Price from £50.00 to £80.00 - http://www.heathrowtaxis.org

There are usually taxis queuing for customers at London's airports.  Ask the driver or minicab company how much your journey will cost beforehand.  Only use a black cab or reputable minicab and never use unauthorised drivers.  A metered trip in a black cab to/from Central London generally cost from £50 to £80 and take 30 minutes to 1 hour.  All Heathrow Terminals have an approved taxi desk and authorised taxi rank.

From Central London to the World Paper Money Fair?

Have a look at the details on the Public Transport Page

Prices correct as of March 2011