The NYC _LEASED_ the B&A in 1900, and at that point all B&A's physical plant and equipment became NYC and, originally, was lettered NYC&HR. Afterward the B&A essentially became a shell corporation, whose stockholders received a fixed rental payment from NYC. From the beginning, however, the Bostonians were unhappy about "their" railroad being owned by a (gasp!) New York corporation, and I'm told that the first general manager that the NYC&HR installed was rather insensitive about this. So as a PR gesture, the NYC went back to lettering everything "B&A", even though it really didn't need to do so. Said another way, for operating purposes the B&A essentially became a state of mind after 1900. During the late '40s and '50s, though, NYC diesels and larger steam power (mostly L-class Mohawks) began filtering in, and gradually the B&A "identity" was lost. (The new Budd "New England States" trainsets, for example, were lettered for NYC.) There was no fixed date for this, but you could probably say that it essentially ended when the last B&A-lettered steamers disappeared. (But I guess some lettering on structures may still remain, somewhere.) The B&A as a corporation was finally merged into the NYC 4/3/61.
The B&A situation was thus different from some other NYC subsidiaries that were operated as separate entities, with their own equipment -- e.g., the P&LE and P&E. In these cases, the NYC only owned a bare majority of the stock and it cheaper to leave things as they were. (This was also true of the Big Four, MC, and T&OC before 1936.)