The whole story is pretty clearly 'a few years later' but is there at any point where they do give a precise time that happened between games?
Probably 5 years or so, since:
Hudson's Gerudo daughter looks about that age according to the internet, and she wasn't born before the end of Breath of the Wild. Although, she only provides a minimum age, since we don't know when she was born. It's only assumed that she was born right after the end of Breath of the Wild.
Plus, we know what a 12-year-old Geurdo, a.k.a Riju looks like, so we know that Hudson's daughter isn't that old.
Wondercricket has some good points for a lower limit on the time gap, but we can also put an upper limit on the time difference based on a few factors:
Most obviously, almost no adult characters appear to have appreciably aged. We can assume that Hylian aging works similarly to humans (although the fact that Impa and Robbie are both well over 100 years old, without any age-changing magic like Purah used, indicates that their lifespan--at least the shiekah's--is at least bit longer on average), so it's likely not been more than ten or so years (you would expect Link and Zelda, in particular, to look substantially different after that time).
Some child characters also don't seem to have aged much. For instance, Shibo and Darton of Dueling Peaks Stable look largely the same as in Breath of the Wild. This limits the time span more severely--children grow faster than adults after all. At a guess, I would say this limits our time gap to no more than about five years--I could maybe, at a stretch, believe Shibo and Darton being around 5 in BotW and then around 10 in TotK.
There's been some progress, but not too much, on rebuilding Hyrule. The Zora have built Mipha's Court on top of Ploymus Mountain, there's a few more buildings (and a lot more decorative mushrooms) around Hateno, Lookout Landing has been built, and the skyview towers were (apparently) only just finished¹. But much of Hyrule is still in ruins between the settlements, and the castle and Castle Town are in much the same shape as they were. Even assuming the castle and its town were deprioritized in favor of the towns and villages where people actually live (which would be very much in-character for Zelda), there can't have been more than a few years of work here. I'd guess no more than three years, but I could believe four or five. Much more than that and I'd expect to see some progress on Castle Town and people moving back there.
As Malady notes in their answer, there's also the case of Mattison, who wasn't born yet at the end of BotW, and has grown a bit past the "toddler" stage as of TotK. We don't know just how quickly Gerudo children grow, but assuming it's similar to hylians (which as previously noted, we're assuming are similar to humans), this would put her around 4 to 6 years old, meaning it must have been at least that long since BotW.
Note that Purah's changed appearance does not give us any time information, as it's revealed in one of her diaries that she was experimenting with age-changing magic again.
¹As an aside, I'm not sure why the skyview towers were prioritized over fixing up the castle and Castle Town. They seem like a major undertaking, and not one of particular importance. Sure, they're important for getting to the sky and for being the map-uncovering gameplay mechanic, but the sky islands weren't there when the towers were being built, and having a map is less important, I would think, than fixing up the still-ruined castle town. This is one I can only chalk up to "Purah and Robbie are mad scientists in every sense of the term".
As Pinckerman stated in the comments, there is currently no canonical answer to how long Tears of the Kingdom takes place after Breath of the Wild. Based on various elements in the game within, the general consensus from various sources is that Tears of the Kingdom takes place anywhere between 3 - 5 years, some even might say up to 10 years.
Personally, two major factors in the time gap estimates are:
Civilization has drastically improved, namely the Lookout Landing. In Breath of the Wild, it was merely the ruins a hylian town and now it is a fully functional town. Even with the technology in the game, this type of development would take some time.
Tulin, who made an appearance in Breath of the Wild as a fletching, is now on his way to becoming a Rito warrior. We don't know how the Ritos age, but it's obvious that there is a drastic age difference even if they did age like humans.