I love Daniel Jones’ Everyman’s English Pronouncing Dictionary for many reasons, foremost among them its scrupulous care in distinguishing the various contexts in which the word might be found (“Note.—Earl Waldegrave is ‘wɔ:lgreiv ['wɔl-]. Some others with this name pronounce ‘wɔ:ldǝgreiv ['wɔl-], In Waldegrave Hall the pronunciation is ‘wɔ:ldǝgreiv ['wɔl-].”) and especially its cheerfully verbose guides to local usage. S.v. Waltham:
Note.—The traditional local pronunciation at Great Waltham and Little Waltham in Essex is ‘wɔ:ltǝm, and this is the pronunciation used by those who have lived there for a long time. Some new residents pronounce -lθǝm. In telephoning to these places from a distance it is advisable to pronounce -lθǝm; otherwise the caller is liable to be given Walton(-on-the-Naze), which is in the same county.
No mention, of course, of the pronunciation used in the Massachusetts town, which is ‘wɔlθæm; in the words of the Wikipedia article, “The second vowel is pronounced properly (“Wall-tham”, to rhyme with tall-ham, IPA: /ˈwɔlθæm/), and not elided into a schwa (“Wall-thumb”, IPA /ˈwɔlθəm/) as might be expected in American English.”
N.b.: I have the 13th edition, 1967 (photo here for the time being); I regret to report that the current edition has brutally cropped the entries, eliminating all the personal tidbits that make mine so delightful.