So this is my first time posting, I've been checking around but I'm not too sure on what might have happened.

I just replaced the exhaust on my '98 Disco I and now my truck is overheating. The day that I put on the new one, I took it for a ride and within 15 minutes I as almost in the red!

Their was an leak on the old exhaust and the muffler started rusting out. It also didn't have the tailpipe muffler on it. Did I get the wrong exhaust kit for it or is it something else?

 

It is not the exhaust causing your overheating issue.
You need to check the coolant level, then check belt routing, fan clutch, t-stat, water pump...rent or buy a live data OBDII scanner and see how hot it is actually getting.