If you look closely at the output of replace, you see that there is nothing that matches the form:
Exp[x_ + y_]
Instead, you only see "E" to the power of "some product". Yes this might be a product of sums, but Mathematica will not expand out the products unless it is told to do so.
You may consider using Expand first:
exponent = Expand[(-(((1 + r) (t - t1))/(2 Q tau)) - t/tau1 - t1/tau1)]
You can then run:
Or you may consider building Expand into your definition of replace.