Not sure if you’ve been Herne Hill Velodrome is great for Track time - it’s cheaper than Lee Valley and there are plenty of decent sessions - race training on Monday 7-9pm and Saturday 10 - 12pm and track league on Wednesday evening.
Everyone there is really helpful.
Rental bikes aren’t the best but fast riders do fine on them! Many people ride track bikes there and take the front brake off.
Maybe see you there !