FlightConnections.com lists flight times rounded off to the nearest 5 minutes. Limiting oneself to connections listed on this website as having a flight time of 2h00m or less (as of December 2017), one can construct the following 17-segment itinerary from London to Tokyo, via Shanghai:
- London–Frankfurt: 1h40m
- Frankfurt–Belgrade: 1h50m
- Belgrade–Istanbul: 1h45m
- Istanbul–Beirut: 1h50m
- Beirut–Baghdad: 1h30m
- Baghdad–Bahrain: 1h50m
- Bahrain–Muscat: 1h30m
- Muscat–Karachi: 1h55m
- Karachi–Lahore: 1h45m
- Lahore–Delhi: 1h25m
- Delhi–Kathmandu: 1h35m
- Kathmandu–Lhasa: 1h25m
- Lhasa–Chengdu: 2h00m
- Chengdu–Changsha: 1h50m
- Changsha–Shanghai: 2h00m
- Shanghai–Fukuoka: 1h25m
- Fukuoka–Tokyo: 1h45m
It also appears to be possible to make it to Sydney in 27 segments. Proceed to Chengdu as above, then:
- Chengdu–Kunming: 1h30m
- Kunming–Hanoi: 1h25m
- Hanoi–Phnom Penh: 1h40m
- Phnom Penh–Kuala Lumpur: 1h50m
- Kuala Lumpur–Jakarta: 2h00m
- Jakarta–Surabaya: 1h15m
- Surabaya–Denpasar: 1h10m
- Denapasar–Dili: 1h45m
- Dili–Darwin: 1h20m
- Darwin–Gove: 1h40m
- Gove–Cairns: 1h45m
- Cairns–Townsville: 0h55m
- Townsville–Brisbane: 1h50m
- Brisbane–Sydney: 1h35m
Finally, London–Cape Town can be done in 15 segments. Proceed to Belgrade as above, and then take the following flights:
- Belgrade–Athens: 1h30m
- Athens–Cairo: 1h50m
- Cairo–Medina: 1h50m
- Medina–Khartoum: 2h00m
- Khartoum–Juba: 1h00m (though I suspect that this should be 2h00m, given that the return flight is 3h00m)
- Juba–Nairobi: 1h40m
- Nairobi–Dar es Salaam: 1h25m
- Dar es Salaam–Pemba: 1h10m
- Pemba–Nampula: 0h40m
- Nampula–Beira: 1h15m
- Beira–Johannesburg: 1h50m
- Johannesburg–Bloemfontein: 1h05m
- Bloemfontein–Cape Town: 1h40m
Given the inefficiency of the route I found through Mozambique, there may be a route to Cape Town with fewer segments via western Africa rather than eastern Africa.
All of these itineraries are presented with the caveat that I have not verified the flight times with the airlines in question; indeed, I am not 100% convinced that Flightconnections.com is handling the time zones correctly for some of the flights in Africa.