Bikes are allowed on HEX. From the conditions of carriage:
Bicycles are carried subject to these conditions of carriage. They are not carried on trains leaving Heathrow between 7.30 and 10.00 or Paddington between 16.30 and 19.00 except on Saturdays and Sundays.
There is also provision on HEX to buy a second seat for large baggage.
To add a few more resource details, here is the HAL site for cyclists:
https://www.heathrow.com/transport-a...ctions/cycling
and I've attached HAL's PDF of the cycle routes to Heathrow.