In my area (central California), the electric utility does the connection to the service entrance (mast). They will only do this connection after the building inspector has signed-off on the permit.
It doesn't matter how many conductors there are in the existing service, you will need to replace all of this anyway. 200 amp service requires much larger wires than 60 amp, plus a larger conduit and service head for the mast. The requirements are clearly stated in the National Electric Code (or whatever code applies to your area).