Realistically, what is the oldest person an agency is likely to hire? Assume that the candidate is exceptionally fit and scored very well in the testing phase.

Probably about 45.

Many agencies would hire someone past the age of 45 if he or she is a good candidate. All else being equal, age doesn't matter so much. Actually I would rather hire someone a bit older than someone too young!

I became a police officer when I was 40.

I became a police officer at age 49 after retiring from the Navy.