Definitely do not take the Heathrow Express. The Piccadilly line is much cheaper and easier, and probably quicker for this journey. You should have no problems getting a seat and space for your luggage as the trains start at Heathrow. Otherwise, London is a large and crowded city. It is busy at times and you may have to stand on public transport even far outside rush hours. If you feel this is a problem you can always ask someone to give you a seat. Whether they do so is up to them.

Mind the gap...

The Night Tube only runs on Friday nights (that is, early Saturday morning) and Saturday nights (early Sunday morning). So it depends what you mean by "3 am on a Friday". At 3 am, the Night Tube is busy, mostly full of drunk people coming out of clubs and pubs. The majority are harmless and even friendly, but you may encounter the odd person who is urinating in a station or singing loudly to himself. Personally I would be cautious about exposing a three year old to this, but that is a matter for you of course.

The underground is not a very exciting introduction to London, and with the early departure time I would suggest you take up the airport transfer options, by hire car (taxi).

Note, however, that the Piccadilly line is not really underground for most of its run between Heathrow and Earl's Court -- it's on the surface all the way from Hounslow West to Barons Court, so it's more exciting than an all-tunnel ride would be.

Uber is extremely cheap to get around London with if you don't like trains :)

I'd mention that many stops can be faster to walk than take the metro; especially in zone 1. If you know any of your trips in advance, it's worth quickly checking that walking isn't a better option.

@WeatherVane: If you think the underground is not a very exciting introduction to London, you are clearly not a three-year-old :-)

Trasport for London has a map showing walking times between stations: content.tfl.gov.uk/walking-tube-map.pdf. That said, if your 3-year-old is anything like mine "walking times" can be anything from 1 to 100 times longer than estimated :D

Not only is he not into walking in crowd he is quite heavy as well. Although I'm fit enough to carry him to places I don't know how quickly it will become tiring :D But I do look forward to it. Good time to shed some pounds

My guess is Junior will be talking about the double-decker (tourist-oriented) bus and the Underground long after every other part of the vacation is forgotten. (OK, maybe not the Crown Jewels if you go.) BTW, I know 5 is a lot older than 3 in a way, but my then 5-year-old spent well over an hour fascinated in the London Transport Museum.

Taking an uber might be around the same price if you are more than 4-5 people. Definitely go for the piccadilly line.

@nickpick not when it is busy. Uber will then charge up to 3 times the usual rate. And OP + wife + child is not "more than 4-5 people".

@HankyPanky if Junior is heavy, please be aware that many underground stations in central London are very deep, and have several long escalators to get down/up. That might cause difficulties for a stroller, or carrying a child. As mentioned, other people in a hurry like to walk past others on escalators. There is usually a lift (elevator) for wheelchair users.

I'm doing the trip once a week and from Heathrow it's very rare that there is a surcharge.

"Very rare" so you have been hit by a surcharge!

@WeatherVane They are quite infrequent to be honest. Is this happening to you a lot?

@Calchas no, I do not promote Uber cabs, but obviously, it has happened to you. Mostly, their "cheap pricing" is marketing BS, where I live (not London). Usually they are more expensive than established local companies, and sometimes 2 or 3 times the usual price, according to local taxi-takers.

@WeatherVane Ah right. Compared to black cabs, they are a lot cheaper, even during surge pricing. Personally I'm not too fussed about price, it is more the ease and convenience. I do use other taxi summoning apps as well. :)

There's no such thing as a "bonus question". If you want to ask separate questions, post them as separate questions.

Sure - Bonus questions removed

If fthere is a surcharge, it will be less than 10%, definitely not 3x. That only happens in central London on New Years Eve. From heathrow you can assume there will be no surcharge. Alternatively you can use minicabit.com to get the best offer when preordering a cab, but the cheapest will almost always be more expensive than uber, especially for pickup from the airport, and if you insist on taking the shortest route, rather than the quickest (that is especially important when coming from Gatwick, less so from Heathrow).