5 Months is a really long time. There is no way that a repair should take this long.
I would have assumed within the first couple of weeks you'd know if it was something that could be repaired or needed to be sent to Omega.
I can't understand why for a watch that isnt that old, it's so hard to repair/source parts. Is it a watch that is still in production/sale?
If it was me, I'd ask for some sort of tracking reference or phone Omega direct to ensure they've received it.