The lifts in A down to the trains have a lower level labelled staff only even though it's not officially staff only. I've checked that with many people to ensure that it's really ok for us to use it as I don't want to miss my flight due to security detaining me thinking I'm up to something
As part of the improvements to the area escalators are due to be added as well as moving walkways.

Once in the tunnel it's a simple case of walking until you see signs for B if that's where you're going. In the case of the C gates the tunnel actually rises up to meet the rail platforms.

I prefer this route but do not do it if you have a lot of baggage or if you don't want to walk long distances.