London's airports are spread around the city and are not linked by a dedicated airport rail network. Changing airports can take a substantial part of a day, especially when immigration, baggage collection and a second security check are involved.
Direct coaches
National Express runs direct coach services on several airport pairs, including routes involving Heathrow, Gatwick, Stansted and Luton. A direct coach can be simpler with luggage than taking trains across central London, but road traffic makes the timing less predictable.
Rail usually means crossing London
There is no single rail service connecting all London airports. Many rail transfers require a change in central London and sometimes a change between stations or transport systems. This can be efficient for light luggage, but it is a poor plan with a tight connection.
Leave a large buffer
An airport change is not an ordinary flight connection. Allow for immigration, baggage reclaim, the surface journey, check-in or bag drop, security and disruption. If the itinerary is on separate tickets, the second airline will normally treat a missed flight as your problem rather than a protected connection.