The little stream that flows under Harrogate Road and under Alwoodley Lane, then into Wigton Knowle, does flow into a drain which goes around the eastern side of the reservoir.
I've spent a lot of time at Eccup during the summer and managed to trace some of the little streams in the area. I used a Philip's street atlas of West Yorkshire as a map, so all place references can be found there.
From the map, the stream that runs under Harrogate Road, Alwoodley Lane and into Wigton Knowle, appears to drain into the reservoir at the deep inlet at the south-east corner. However, it drains into a small culvert, hidden by trees, a few yards short of the apex of the inlet.
Once underground, it is joined, on the left, by another culvert which surfaces to the west of the inlet. This stream can be traced, on the surface, as far west as the area near Reservoir Lodge. I haven't followed this stream to its source as it flows into the Reservoir Lodge gardens.
The culvert follows the eastern edge of the reservoir and somewhere close to the gatehouse at the end of the lane leading down to the reservoir from Harrogate Road. The waters from the culvert emerges at the top of a magnificent flight of 142 steps and joins Eccup Beck.
I'm surprised that the culvert outfall hasn't been mentioned before, but it is hidden just inside the treeline. For completeness, I looked at the drainage of the north side of the reservoir. Eccup Beck runs in an open ditch along the north side of the reservoir with a very short modern culvert which takes it under the road across the embankment into a steep open channel which runs diagonally down the embankment.
I've put together a small website with some photographs at
http://www.urbanredux.uk This is fantastic - I've meant for ages to go up to Wigton Knowle and find out what happens to this beck, but just haven't had chance. Good to finally confirm what I suspected, ie that it doesn't flow into the reservoir but gets diverted around it, albeit in the opposite direction to what I'd assumed.