Brick
Lane still exists, but none of the other streets referred to in the
directory listing do, since the area was rebuilt after The Blitz
flattened that bit of London.

Up to a point. There are other streets near there which have kept their
names. Eg Sclater Street (which is sort of the continuation of Hare
Street on the West side of Brick Lane and is listed in the 1921
Directory). Ditto Bethnal Green Road. And while the blitz did a great
deal of damage there are a fair few buildings surviving nearby,
including in Cheshire Street the mildly infamous Carpenter's Arms.

Hare Street apparently sort of followed
the current Cheshire Street's course, with No. 80 being not far from
the existing Three Colts Corner.

Yep - Hare St was what is now the Western part of Cheshire Street - as
the OP can see on the Bartholomew's of the time thanks to
http://www.hipkiss.org/data/maps/bar...0_1473_600.jpg

--
Robin
reply to address is (meant to be) valid