There are two things to remember about these signs.
The first is that no one can stop you walking on a public right-of-way e.g. a public footpath. If you find a path blocked please contact Wiltshire Council with a copy to me.
The second is that trespass is a civil, not a criminal, offfence and so the police are unlikely to take any interest in reports of people walking on fields. Also, it would be very dificult for anyone to prove criminal damage to grass.