The most logical route which covers the maximum on foot, although not in Piccadilly Line order, would be to walk from Hatton Cross to T5, then from T5 to the top of the tunnel. It would then be necessary to take a TfL bus through the tunnel, before completing the walk to the T1/2/3 platforms from the Central Bus Station.

I would guess you can fiddle T4 in there somehow.

There is no foot access to the Tunnels now, as you've been told.