uo doesn't stand for [øː] though, it stands for [uo]. uo developed in Old Dutch and Old High German from Proto-Germanic stressed ō in all words (OHG has guot for *gōdaz), and in a few also from wō (kuo (“cow”) < *kwō-, Old Dutch huo < *hwō). There is also a parallel change from ē to ie. I think the earliest Old High German texts have ua or oa, but most have uo. I think they also have ea and ia early on, and ie later. ( has ua and ia) So *muotan certainly came from *mōtan, not *mōtjan. But I'm not sure whether the change ō > oa/ua and ē > ea/ia happened already in Frankish.
Do you know if there is any evidence in Old French regarding this change? We know that Germanic au appears as ō, but what does Germanic ō appear as?
Well, if the same sound ends up in modern French as two different sounds, I wonder why that is. So I'd like to be able to compare other words that were borrowed from Germanic ō, to see if there is any general rule or pattern to them. And maybe through that, we can figure out whether Frankish still had ō, or whether it had already become a diphthong.