A reamer is the best bet. You could even start off with the careful application of a countersinking drill bit. Care is obviously the watchword...because you want the hole in the middle !
There's usually at least a ridge or step of some kind to indicate the area of the "jukebox" hole. Very many records are pressed with slightly off centre holes...so don't get too paranoid about it. The main thing is be sure that you don't damage the playing surface with the debris or any tool you choose to use.
Yes, many years ago now. A mate of mine acquired a Juke Box and wanted to load a number of singles without the partially cut large spindle hole.
It's one of those jobs where the tool is perhaps not the most imnportant thing. It's 90% care and attention...and shouldn't be attempted by the habitually "ham fisted" ;^)