Any question answered...

Thanks - I've just cut a length of wood to 202cm which obviously hasn't the depth as the bookcase to test the diagonal but I've discovered that where the thing will go there is actually 4cm space to the ceiling. I don't understand your calculation enough to apply it to what is now a ceiling height of 206cm but I guess it makes it slightly easier.

Pythagoras theorem.

A right angled triangle with sides of 202cm and 28cm (the depth of the bookcase) will have a diagonal length of 203.91cm.

a^2 = b^2 + c^2

b=202 (height)
c=28 (depth)

a = sqrt( b^2 + c^2 ) = sqrt( 202^2 + 28^2) = 203.91cm

A ceiling height changing from 204cm to 206cm doesn't change the calculations, but having 206cm ceilings means you'll have ~2cm of clearance when you tip the bookshelf up from horizontal to vertical.

tl;dr you'll be fine