Removing/re-seating threaded rod for cabinet leg
Technically that is threaded rod not a threaded insert. A threaded insert would be an insert with threads to accept a threaded rod or machine screw. 

Get two nuts that fit the threads, thread the first nut on far enough that you can get the second nut on.  Using two  open ended wrenches  ( or adjustable wrenches will work ) tighten one nut up very tight against the other, I.E. turn the bottom nut counter clockwise as you turn the top nut clockwise. This will lock them together, now you can use one wrench on the bottom nut to try and back out the threaded rod. The two nuts locked will not back off the rod it will try to turn the rod. 


  I have tried getting my pair of needle nose pliers on it as they are
  the best thing I currently have to hand, but they just appear to slip
  off.


As you have found out, needle nose pliers are not the correct tool for removing nuts, they will not give you enough grip to accomplish this task.

It may be glued in or very tight so you may need to secure the leg in vise or clamp it to a table top, protect it by wrapping it with something. 


  I'm assuming the pilot hole will be drilled at right angles.


If the rod is at an angle ( not the correct angle ) then the hole itself is at an angle. 

Once/if you get it out you will need to fill the hole with a dowel and glue so you can drill a new straight hole or put in a threaded insert, sized correctly, to accept the threaded rod.