score:7
I'll partly answer both of your questions at once - I'd suggest getting the Picadilly line from the airport in. As the trains start at the airport, you'll be getting on an empty train, which means you can be sure of getting seats and having space for your luggage (there are spaces by the main doors of the trains for suitcases).
You'll also then be able to introduce your son to the concept and feel of an Underground train while it's still fairly quiet rather than jumping straight into a busy city-centre service, which should help him to acclimatise to it.
For the bus, I'd suggest doing something similar - get on a bus at it's first stop, while it's empty, and pick the upstairs front seat - he'll be too engrossed in the view to be scared! The tourist routes often run shorter journeys starting and finishing in the city, which can be good for this - e.g. Trafalgar Square to Tower Hill.
Upvote:3
All above answers are correct in their remarks. And you can even receive realtime information to the millisecond in your smartphone, and program an apple-watch to vibrate at the optimum time to board a train.
However you can also go for the zen-way and just chill out and enjoy your family trip to London. Stay away from the tubes & buses on rush hour, and just enter the subway when you need it. If a train is terribly crowded, wait for next one and let your three year old amaze at the beat of the society.
Upvote:4
Bonus 2
Here is a map showing night tube lines which includes Picadilly line between Earls Court and Heathrow. But 3 a.m. Friday is not "Friday night" so I checked further.
I tried to plan your journey: 3 a.m. on Friday by underground from Earls Court to Heathrow, and it advised the earliest train to leave at 5:21 a.m. However for Saturday 3 a.m. the advised train departs at 2:55 a.m. So sadly, the answer seems to be NO.
Other matters:
If believe there is a bus route app available which helps you to find routes and stops.
Don't overlook the river buses, I think you can get to Hampton Court by river bus.
Upvote:5
Although the tube is extremely busy in the morning in rush hour (about 7am until 9.30am), and quite busy in the evenings (about 5pm-7pm), when services are running normally things get a lot less busy outside that window, and some seats will usually available (although not necessarily three together). Unfortunately most lines have cars which don't have passenger accessible connections, which means that if the carriage you're waiting at happens to be full, you can't walk down the train to find an empty one (the new District and Circle line trains do allow this, as well as having working cooling systems, on the other hand it's not the more reliable service in the world). With enough time, it's possible to learn which bits of a train tend to be busy at a particular station, but that's probably not useful for you.
In general, things are a bit easier if you're heading out of central London in the mornings, and into town in the evenings, only crossing the centre near the middle of the day, but that would limit you to West London to a certain extent. If you have a lot of free time (i.e. over an hour) then the London Overground forms an "outer loop" which would usually be significantly less crowded than a trip through the centre, but obviously slower.
Finally, the Tube map has won awards for its clear design, but it lies when it comes to how close different stations are to each other. If you're making a short trip in the centre, check a street map, or else you can end up walking further to reach the platforms than you would to get there on foot above ground.
In terms of not annoying the locals: