Are there nonstop flights from Tallinn (TLL) to London (LGW)?

Yes. AirBaltic fly this route. Flight time is about 2h 38m.

What is the fastest connection from Tallinn (TLL) to London (LGW)?

via Copenhagen ≈ 3h 8m, via Amsterdam ≈ 3h 10m, via Hamburg ≈ 3h 10m (estimated flying time, excluding layovers).

from 15,000 ANA miles (economy, one way, based on the published award charts).

Lennart Meri Tallinn Airport (TLL) to London Gatwick Airport (LGW)・1,816 km

Nonstop flights available (1 airline). Flight time about 2h 38m. Nonstop return flights also operate.

With milesANA 15,000 miles+ (via ZRH)Economy, one way, estimate. Tap a connection for miles by cabin.

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27 one-stop options, sorted by estimated flying time. Tap one to see the airlines on each leg and booking links.

Times are estimated from distance. Check schedules, frequency and availability with the airline or a booking site.