How do you know it was not private? Many bridleways pass through private land and are not automatically a right of way. The landowner can ask you to leave at any time.
That’s wrong. 100% incorrect.
If you are on a waymarked public right if way such as a bridleway or footpath then you have the right to cross and recross, even if that RoW is on private land. There are of course caveats around keeping dogs on leads, not straying from the path, closing all gates etc but you can’t be asked to leave a public RoW, nor can it be blocked by the landowner.
If it needs to be blocked (eg for major groundworks), then the landowner needs to apply for a temporary restriction from the council/Highways Authority.
If you’re off route, then yes, it’s reasonable for the landowner to ask you to rejoin the correct path because you’re no longer on de facto public land, you’re on private land.
There’s a whole range of requirements that fall on the landowner not to keep dangerous animals in fields through which a public RoW passes, not to plough over a RoW, not to spray pesticides over it.
Report to Highways Authority and National Park.