From the West End, take the Piccadilly line; estimate around 50 minutes to Terminal 5

https://www.scribd.com/document/1074920514/Getting-to-Heathrow-from-Gatwick-Easiest-Inter-Airport-Transfers-178501

From the West End, take the Piccadilly line; estimate around 50 minutes to Terminal 5, and consider traveling off-peak for a more comfortable journey.