I think blue plaques are generally the property of either the National Trust or English Heritage. They are fixtures, not a fittings. Why does that matter? Well, the original owner of the property would have to have provided consent for the plaque to be erected and possibly planning permission would have had to have been granted by the local planning authority (in which case permission should have been sought to remove it). The plaque itself may still therefore belong to the NT/EH - even after being added to the building as it as a fixture, not a fitting. We don't even know who this guy is in the photo, but if the owner removed the plaque, then they might be required to return it to the care care of NT/EH, since it might not be theirs to sell. The same could apply to subsequent purchasers of the plaque.
Clarification:
EH provides the following guidance:
https://www.english-heritage.org.uk...plaque/commemorative-plaques-guidance-pt1.pdf
In addition, some choose to clarify the future ownership of the plaque itself. For instance, the guidelines in use by Aberdeen City Council state that the Council ‘will retain ownership of the plaque’, while those of Guernsey’s Blue Plaque scheme state that ‘The plaque will belong to the Museum [i.e. the administrator of the scheme] after its fixture to the building’. Care should be taken in making such statements, and they should ideally be based on legal advice. As is discussed elsewhere in this document (see pp. 99-100), it is usual to find that – once a plaque has been installed – it is viewed as having become part of the property to which it is affixed.
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Questions need to be asked: who legally owns the plaque? was planning permission required to erect it? If so, was permission granted to remove it?