Heathrow Connect is much cheaper than Heathrow Express. My advice would be to avoid Heathrow Express if budget is a consideration.
It also stops more, so you can get to Maidenhead without going into central London, changing at Hayes and Harlington. Still £33.70 per person, which seems steep.
https://traintimes.org.uk/heathrow+terminal+2/maidenhead/10:00/2017-07-25 - like
@vickster says, a people carrier taxi or minicab booked from Maidenhead may be better.
The Bike Box Alan (92cm x 113cm x 32cm) may be great for flying but it's too big to take as standard luggage (30 x 70 x 90 cm) on UK trains, at least officially, but as long as you don't block aisles or used doorways, I'd doubt anyone would object - anyone tried in practice?
If you've nothing else to do, storing the box at the airport and one person taking the suitcase by train and circle line tube while the other cycles might be the nicest way to get everyone and everything to Tower Bridge area: NCR4 is one option and probably has some pretty bits and some rubbish in its 45 miles, although NCR61 and the A4 cycle track looks about 10 miles shorter. I suspect locals may know better routes.
Easiest might be to take the bike, unboxed, off-peak on a stopping train to Paddington, head for Hyde Park and then follow CS3 to the Tower Bridge area.
Or can you unpack the bike and leave the bike box at left luggage at Heathrow if such exists?
It exists but it ain't cheap:
http://www.heathrow.com/airport-guide/terminal-facilities-and-services/left-baggage