The small-bore side tunnels were built for such traffic. One was for pedestrians (in the 1950s presumably quite a few Heathrow workers walked in from buses on the A4) and one for bicycles.
A few years ago, as usage had become negligible, they were converted into supplementary entry/exit tunnels for cars. As part of the deal for abandoning pedestrian right-of-way access, regular London Bus services were made free on the section from Heathrow to the first stop beyond the tunnel. So the original poster can take any bus from Heathrow out to the A4, then walk the rest of the way, for nothing. BAA make an annual payment to London Buses for this service.
BTW our house is a 20-minute walk across the docks from London City terminal and I always like to walk to and fro, if only as a discussion point at my destination meeting !