The only thing I can do is tell you that, according to Google Earth, for instance Goulston Street was/is about 259 yards long, and Osborn Street about 144 yards. From there you should be able to work out the scale of whatever map you have/use.
"You can rob me, you can starve me and you can beat me and you can kill me. Just don't bore me."
Clint Eastwood as Gunny in "Heartbreak Ridge"
Perhaps the easiest & most accurate way is to use the scale at the bottom of the map.
The larger OS maps come in different scales (1:1056, 1:2500, etc) - which in turn have been rescaled in various proportions according to the reprint you have.
So use either the map scale or, if you're working off a digital copy that doesn't have this (bad), a known measurement (such as street/building width) to establish scale. Tower Hamlets planning register will give you plenty of hard readings.
alangodfreymaps.co. uk sell excellent paper reprints dirt cheap. They come with the scale, too.....
If you're trying to measure smaller distances of past landscapes, Google Earth will serve as a rough ballpark tool. Besides changes in street lineages etc there is some parallax distortion in the satellite images themselves.
Together these work reasonably well. (Eh, Mr Blue? )