Edit. Of course, I would find the mistake after posting.  k ranges over the even numbers from 2 to 2h, so the result is not as nice.  I will try rescuing the approach.   End Edit

Each term is close to (m- k+ sqrt(d) + 1/2)(m - k-  sqrt(d) + 1/2), where k ranges from 1 to h=(m-n)/2.  (h may be off by 1.) Your product is close to n! squared times the product of two binomial coefficients: m+ a choose h and m-  b choose h, where a anb are close to sqrt(d).  Letting a and b range over floor(sqrt(d)) and ceil(sqrt(d)) are a start if h is not too small.
For finer estimates, let a and b be appropriate real numbers.

Gerhard "Email Me About System Design" Paseman, 2011.06.21