I don't know if that hadith is really said by our prophet, or what it really means, I will talk about how I understand Allah and time, based on Quran and try to provide an opinion, rather than a solid answer to your question, as noone can provide it because noone knows what Allah is doing at a given time, other than always seeing and hearing us and being close to us (Verse 50:16).
There is Allah, a being that is not created, a being that exists indefinitely, with no start and with no end, without a need, with no equivalent (Surah 112). There is everything else, created by Allah, all need Allah for their existence; the universe we live in, space, stars, planets, earth, angels, humans, jinn, animals, plants, other possible universes and creations we don't know of. From the creation I have witnessed so far, the soul, the intelligence and the free will are among the top of the creation in terms of their beauty and complexity.
As the creator, my understanding is that Allah is not "in" the creation. Again, in my understanding, all the creation is a part of Allah, an extension of Allah, and always controlled, monitored and logged by Allah (Verse 72:28). I think about it like an online world, where everything is hosted on a server (Verse 27:75), all the creation, all the events, all characters (humans and all others) and their actions are being logged (Verse 18:49). Allah does as It wills, when and the way It wills (Verse 3:47). There are things about Allah and the creation we (humans) can know and understand to an extent and there are things that we can never know and understand.
Now, let's focus on time.
I consider time as the flow. Flow of actions, flow of life, flow of planets around their orbits, flow of rivers, flow of thoughts, etc. Every flow has a start and an end. Between the start and end, or between the different points of that flow, one can make a measurement, even though that measurement may differ from being to being. How a human perceives the flow (the time) may be different than how an animal or a plant or any other being perceives it.
Allah has no start and no end, as I understand it. Hence, there is no flow for Allah in the same manner there is for us. At least not in the exact way that we perceive, measure and record (Verses 10:45, 17:52, 22:47, 23:112-114, 32:5, 70:4).
As you implied, when it is late night for us, it may be morning, noon, afternoon or evening for others on the other sides of the Earth, but that is how we perceive and measure the time as human beings. With the light of the things I explained above, I think of Allah to exist anywhere and everywhere, any time and every time.