How important is it to avoid nicking the center core of a coax cable?
The answer depends on the extent of the damage caused by the nick and where it is.  Severe damage to the center core can result in noticeable signal degradation.  In analog service applications, this can cause all sorts of weird visual distortions (e.g. vertical lines, dash lines, random dots, etc).  In digital service applications this is a bit more dramatic (e.g. drops in cable internet speed, voip service skips, etc).  

Without professional tools to verify signal strength and clarity, it's going to be difficult to say for certain.  I'd venture to guess that if you don't notice any degradation in video quality, you're probably OK. 

If you're going to be doing this multiple times, you might want to invest in cabling tools (e.g. crimpers, coax strippers, etc).  They can make the job a LOT easier and reduce the likelihood of nicks occurring. 

There's also a good set of instructions here showing the different cabling technique for various coax connectors. 

-M