hi gatty7 . even i can see the chains are dry , large sprocket on the reel is BADLY worn so yes the chain will jump off . its looks as though it unevenly worn too so even worse !
as all know i am not a reel mower guy (too much like hard work for me) but i did use when i hadto service one or two was TAC2 on the chains , a good sticky lube simillar to bike chain lube but i did use it on hoist posts when i was on the tools and stayed in place for ages easy to clean too .
roller is way too worn to be of use now . BB can tell you for sure or deejay too but maybe the block type of thing ? replaced the rolloer and far as i know still readily available too . just sort of a rubbing block not a roller but still needs to be lubed well and cleaned often .
sorry for jumping in but even im not a 45 fan i can see some problems there

, Dave