I don't think there's any explanation per se, they just proceeded as if it wasn't an issue or had resolved. Sometimes movie makers gloss over these details assuming the audience will come to their own conclusion. I figured, since his return to Downton was only temporary, it wouldn't be an issue short term. Long term would have probably been a no-go and if that had happened, I'm sure they would have mentioned or addressed the palsy he had suffered from at the end of the series. Good question!
I think since he had been resting, it enabled him to do his job as Butler for a few days without much worry. I think his disease is sort of "on and off" so to speak like with good days and bad days. So the events of the movie must've been on his good days