I don’t race and I don’t live in either of those places, but….
Herne Hill is the track in London. However, it’s outdoor, so only open for a limited amount of the year. I understand there is plenty of riding and racing that takes place there.
Info here: http://en.wikipedia.org/wiki/Herne_Hill_Velodrome
Not sure which Newcastle you mean (on Tyne or under Lyme)?
List of UK tracks (indoor and outdoor) here: http://www.roadcyclinguk.com/news/article/mps/UAN/191/V/2/SP/