The hotel in "Bexhill" at which Poirot and Hastings stay is actually the Royal Victoria hotel in St. Leonards-on-Sea, just east of Bexhill.
With the help of Joan I found a nice number of locations for this episode. I think the racecourse has been filmed on at least two different locations, but where? Here are my results so far: commons.wikimedia.org/wiki/Category:Agatha_Christie_films. Look for the link to The ABC Murders. A direct link to this site doesn't work on this board because of the ) at the end. Enjoy!