I think you are right about Lambton (or Ayton) not being completed until the 80s. The route for the 638 between the Galleries/Town Centre Bus Station and Wear Industrial Estate doesn't mention any additional roads between those destinations in both the 1976 and 1979 PTE guides so I'm presuming the route operated direct via A1231/A195 to the Mackro interchange.

Buses from Washington to Chester (180, 181, 531, & 532) all operate northernly via the Mill House Roundabout and Portobello (181 goes via Wrekenton) as opposed to southerly via the Lambton and Ayton route. Washington Town Services (177/178) serve every other area of Washington except from Lambton and Ayton. By 1985, buses are going to through Lambton Village Centre but it'll be interesting what the 1982/3 timetables show.