Do overs bowled in training and nets count towards the fast bowling limits?

Yes. Under the ECB Recreational Cricket Safety Regulations, a delivery counts towards the spell and daily limits when it is bowled in a match or in training with the intention of bowling at match or near-match intensity.

Deliveries clearly meant to be slower do not count, for example walk-throughs or running in at half (or less) of normal run-up speed.

The limits apply to hard ball cricket. The ECB's recommended weekly workloads, however, count soft ball and hard ball overs alike, because both put similar forces through the body.