The short answer is that we don't really know for sure but we have some ideas on the subject.
The chief eligibility criterion for English Heritage's scheme is that the building on which the plaque would be mounted is an original, contemporary with the person being commemorated.
No buildings that had a direct association with Haydn are left now. The Hanover Square Rooms site is now occupied by a Japanese bank and the house rented for Haydn by Rebecca Schroeter in Bury St, St James's, has disappeared under a re-development that now houses the Christie’s auction house. St James's Church in Piccadilly, where Haydn witnessed Therese Jansen's marriage to Gaetano Bartolozzi, is still standing, but ecclesiastical buildings may not carry plaques of this sort.
There have apparently been a number of attempts to put up a plaque under an independent scheme (in the same way as we are doing) but it's not clear whether they foundered for reasons of finance, lack of permissions for the chosen site or any other reason.