Walking from Manchester Piccadilly to the Alan Turing Building should take around 20 mins. Leave the station by the Fairfield Street exit (head down the escalators or lift from the main concourse) which brings you out at a big road junction. Cross both main roads, and go along a smaller road (Granby Row) to the left of the Bull's Head pub. Keep straight on, as the road becomes a pedestrian walk and then a road again, and at the phoneboxes turn left onto Sackville Street. Go under the railway bridge and continue under a bridge between buildings, and where the road bends off to the right, follow the left-hand pavement which becomes a footpath and goes through an underpass. Afterwards, keep left under the motorway flyover (avoiding a deeper underpass ahead) before bearing right (avoiding yet another underpass to the left). After very carefully crossing the motorway sliproad, you find yourself on Brook Street. Walk down this (away from the flyover). At the intersection with Grosvenor Street, cross both roads and then continue along (now Upper) Brook Street on the opposite side. Cross the next side-road (Booth Street East, carefully again!), continue past the Aquatics Centre car park and then the Alan Turing Building is on your right. To get into the building, go into the walkway after the second "finger" and then the doors are on your right.
Note that a printer-friendly version of these directions (along with a map) can be found here.