On a flat top, you neither need nor want a new bridge. What we usually do is fill the saddle slot with a new piece of ebony, and rout a new saddle slot. If done right, this is a completly structurally stable and strong process. This is definatly NOT something you want to do yourself, as the fit of the plug is critical. (Of course, a new bridge is not DIY either.)
Take it to someone with experience.
But yes, a new nut.
All of this would run about $150-175 in my shop; $75 for the nut, and $75-100 for the bridge work. And of course, a setup would be part of that (we are, after all, already working on most of the parts which are involved in a setup).
Light
"Cowards can never be moral."
M.K. Gandhi