I have heard of pubs pulling out of darts leagues for this reason.

This may be so - it would be good to have some details. But if LAs are not going to insist on enforcing this - they may be pulling out for no good reason.   

I have seen applications for Regulated Entertainment locally that specify time etc. Are licencees expecting to have to limit the playing of darts etc by their customers in these premises to these times and are the LAs going to enforce this and claim they would be in breach of the legislation if they did not?

I have yet to see an Premises Licence application that specifies permission for pub games.

The Licensing Act 2003 does not (intentionally) charge Weymouth and Portland Borough Council or any other local authority with preventing live music in these premises without entertainment permission....It also does not charge them with the prevention of conventional pub games in these premises...The Act certainly does not charge them with protecting customer's conventional pub games whilst preventing all live music made by customers in these premises.

If the local authorities consider that this is an unintentional charge being placed upon them - by the words of the Act - then they must urgently join in with any attempts to change the situation for live music.