How does someone remove a watch link if the link where the pin has to be reinserted in has no hole?

One takes such things to an experienced technician to have the work done.

Instructions usually come with the watch or wristband. Where there is no hole in the ends of the links it sometimes requires a very thin "shim" tool to be used between the links to pop the spring loaded bar - but as I said every design can be different. If it's a decent watch then it's probably cheaper to have a high street battery replacement kiosk make the alteration than to buy the kit to do it yourself!