I went through here a few weeks ago - I just checked in JOSM and this seems to match reality (with the exception of the artistic license on micro-mapping of the stairs, which obviously don't spiral inwards).

The lifts only have one door, so you have to go around 180 degrees at ground level to get into the lift, then it's straight out into the tunnel at the bottom.

Incidentally, one of the lifts was out of order when I took my bike through - which I think is not an uncommon occurrence - so there's certainly a chance that the cycle route goes up the stairs.