I would want to say something like "exams are just round the corner." , is "Shiken mou chikai desu." acceptable?
Chikai 近い can be used for time definitely.
Shiken(ga) mou chikai desu.
This is a totally acceptable expression, preferably with the particle 'ga'.
Yes, you can use 近い
But you can also use もうすぐ to better express the nuance of "just round the corner".