Wally is a non-canonical character created for for The Case of the Discontented Soldier, episode five of The Agatha Christie Hour.
Wally is Major John Wilbraham's dog, who is waiting for the major outside the local shop in Cobham. Wally appears to be some kind of Spaniel breed.
According to IMDB an uncredited Walter Gaunt is playing Wally. This Walter is also credited as playing the character Wally in the 1976 television series Nobody's House (the series appears to feature a Spaniel dog), also featuring William Gaunt (who portrayed Major Wilbraham in The Agatha Christie Hour) it is not impossible that Walter is the real life dog owned by William Gaunt (they share the same surname, after all!).