Phuket Air briefly offered flights from London Gatwick to Phuket (direct one way, indirect the other) but due to safety issues, the airline was banned from flying within Europe. It then stopped flying altogether in late 2005.
When I visited Phuket in September last year, I flew with Thai Airways from London Heathrow, stopped for around an hour in Bangkok, and then boarded another flight to Phuket. As already stated, flying time is a little over an hour.
Alternatively, have a look at Air Asia and One-Two-Go, who both offer very low fares between Bangkok (both the old and new airports) to Phuket.