I'm a Washignton local if i were doing that commute in summer and dry would be
C2C to, Barmston lane over the 1231, over the traffic lights at nissan way, i think theres a shared used cycle path on the new road there, follow this to the a1290 turn left, turn right at the roundabout onto Sulgrave road, then right on Rutherford road (an alternative would be left then Inkerman road right onto Heworth road left onto Coach road on to Pearth Hall road but i think this will have significantly more traffic at commute times) , left onto Stephenson road, straight on at the roundabout careful this is a busy from A195 traffic , up stone cellar road (bit of a long uphill drag) . right at the 2nd roundabout on to Pearth hall road gets you to the middle of Springwell. Looks to be a muddy bridleway at the end of Belmont terrace then will get you to galloping green road but i'm getting away from places i normally ride.
Disclaimer check it out first im a fairly confident fastish road cyclist.