As Jacinto indicates in a comment above, the original serialized version of Great Expectations— where the extract in question appeared in the December 15, 1860 edition of All the Year Round—capitalizes Time early in the sentence, but lowercases time at the end of the sentence:
"Would you give me the Time?" said the sergeant, addressing himself to Mr. Pumblechook, as to a man whose appreciative powers justified the inference that he was equal to the time.
A search of volume 4 of All the Year Round (October 13, 18690, through March 22, 1861) finds 78 pages containing the term "the time" (sometimes more than once) in the course of the volume—and in 77 of those instances, the term is spelled all-lowercase. The only exception is the sentence that the poster asks about. It therefore appears that the consistent preference of the publication was for "the time" (all lowercase) and that the single instance of "the Time" was probably an error.
The reason that some subsequent editions retained "the Time" in that sentence is that any editor or proofreader who noticed the inconsistency in the original edition must have decided that it was intentional rather than accidental. Legacy errors of this type are a common affliction in publishing.