"I don't watch TV. It's a cultural wasteland filled with inappropriate metaphors and an unrealistic portrayal of life created by the liberal media elite."
It appears to be set immediately after Hayate the Combat Butler, or more likely, just prior to the last chapter. That manga started in 2004 and was set for that year. And the manga covered a single year. Thus I would say it is likely set in 2005 or 2006. But it really is hard to say for sure.
Update: OK, I just saw the raw for the latest chapter (45). Hata-sensei may be playing fast and loose with the timeline.
Nasa gets Tsukasa an iPhone XS, which means chapter 41 has to be set in 2018.