At Southend Airport, you main public transport option is the train station basically outside the terminal. From there, there's a train every 20 minutes for most of the day to London, with stops at Stratford and Liverpool Street, journey time just over an hour.
For Heathrow Airport, you've two options. One is train to Stratford, Central Line to Lancaster Gate, 5 minute walk to Paddington Station (this is much quicker than train to Liverpool Street then Circle Line / H&C Line direct to Paddington), then Heathrow Express on to Heathrow Airport. This would be 1 hour 30 to 1 hour 45, depending on connections and walking speed, cost of about 40 quid without any discounts or railcards. Otherwise, train to Stratford, Central Line to Holborn, and Piccadilly Line to Heathrow. This will be about 2 hour 10 minutes, but only about £17 before any railcard discounts.
For Gatwick Airport, you'd take the train to Stratford, change onto the Jubilee Line, ride that to London Bridge, then take the FCC service to Gatwick Airport. Typical journey time of 1 hour 45 to 2 hours, depending on connections - SEN to London is every 20 minutes, London to LGW is every 15, so timings vary. Cost is £26.70 with no railcard for a single journey.
Heathrow is quicker if you spend more, and generally the HEX is quite a bit more reliable than the FCC services to Gatwick. If you care about cost, Heathrow by tube is quite a bit cheaper than Gatwick by train, and not very much slower than many Gatwick services.
For exact timings and costings, pick your favourite UK rail or tube journey planner!