Does the properness of 'Street' (in the name of a street) survive when discussing two particular streets together?
Solution 1:
In British English, it does survive. In fact, the properness of Street generally precludes the treatment you propose.
An accident at the junction of Queen Elizabeth Street and Tooley Street.
One might possibly hear “Queen Elizabeth and Tooley Streets,” perhaps in a radio traffic report, but since that’s spoken any capitalisation is unclear. We don’t write it that way.