I think the time the LOAs take are just a function of how complicated your case is. From those I know, most single, childless people, who are not eligible for a mobility premium got them fairly quickly.
Also the LOA is not related to the background check. That is done separately and can continue after the LOA is signed. So checking if your references have been contacted doesn't help.
Basically, don't despair if you're still waiting for an LOA. It's just a formality depending on how long HR needs to calculate and approve benefits.