I have just read a circular sent to walk leaders in a national park (NYMNP) that reads as follows:-
"Good morning Walk Leaders,
Please see below message from the Ranger Team, I hope this doesn’t cause any major issues for your walks.
Please avoid moorland walks across open access during the critical bird nesting season (April- July). It’s not something we want to encourage, and the open access rights are likely to be closed so would be trespassing."
Whilst the owner of such Open Access Land can ask what he wants, surely they cannot restrict access like this legally (I believe they can prevent access for up to 28 days per calendar year provided they give notice)