It's talking about 'what a time', or 'what a long time' or 'what a hard time' - as it's describing an arduous journey - note that it is tedious, and that it is lumbering, bumping, jolting, rattling - meaning that the journey seemed long, to the three inhabitents.
It could also mean 'how long was it - it seemed interminable'.
A bit like 'such time!'
The quote continues like this:
What time, the mail-coach lumbered, jolted, rattled, and bumped upon its tedious way, with its three fellow-inscrutables inside. To whom, likewise, the shadows of the night revealed themselves, in the forms their dozing eyes and wandering thoughts suggested.
The 'to whom' contrasts poetically with the 'what time' describing such a long tiresome journey that they were almost losing themselves, becoming almost delirious.
https://www.cliffsnotes.com/literature/t/a-tale-of-two-cities/summary-and-analysis/book-1-chapter-3