It depends on how the recharge is supposed to work. A lot of items like 5e wands are made to have a limited number of uses per adventuring day to represent that they have some power that can be used up and recharges fairly rapidly without any special materials. For those items, tying them directly to rests (long rests in 5e terms, overnight rest in other games) or to simple ticking of a clock works fine. They're not mythically tied to anything, they just recharge their power over some amount of time, and the 'once per day' is really a bookkeeping convenience and not fundamental to how the device works.
How the device works? It works the way the rules say it does. I mean, it's not like there really are magic wands around. In 5e that recharge time may mean "After a long rest". It may not. A person might get powers back after they rest, but since items don't "rest" per se, it's kind of hard to rationalize it that way.
In Pathfinder and 3.* editions, Clerics renew their spells at a "specific time", with examples like "At sunrise" given in the rules.
Player's Handbook page 32 said:
Clerics do not acquire their spells from books or scrolls, nor do they prepare them through study. Instead, they meditate or pray for their spells, receiving them through their own strength of faith or as divine inspiration. Each cleric must choose a time at which he must spend 1 hour each day in quiet contemplation or supplication to regain his daily allotment of spells. Typically, this hour is at dawn or noon for good clerics and at dusk or midnight for evil ones. Time spent resting has no effect on whether a cleric can prepare spells. A cleric may prepare and cast any spell on the cleric spell list (page 183), provided that he can cast spells of that level, but he must choose which spells to prepare during his daily meditation.
(Emphasis mine). So being tied to something like sunrise or sunset isn't arbitrary, and it isn't my invention, it's in the rules and it's pretty specific. Equally specific is that it's
not tied to rest, long, short or otherwise.
Not sure what the existence of 'portable timepieces' has to do with it powers recharging though - a device that needs X time or Y trigger to recharge will just recharge on that cycle as part of the world even if the players don't have a way to tell time with them. The recharge cycle actually might be a handy, if unintentional tool for timekeeping. Also, real world pocket watches were around in Europe from the 1400s, and reasonably common by the 1500s, and self-contained mechanical clocks date back to the 1300s.
The Balance wheel, necessary for any kind of reliable watch, wasn't invented until 1755. Its precursor, the Tompion regulator, was invented in 1675.
Queen Elizabeth was known to have a pocket watch in the mid 1500s, but it operated badly, and was so expensive that only the nobility could afford one. So no, pocket watches weren't "commonly available" in the 1500s, and wouldn't be actually workable for a century after Elizabeth took the throne.
Meanwhile the full plate armor that's routinely seen in D&D only dates back to the 1400s. This means that even ignoring the possibility of magical clocks or gnomes or dwarves have more advanced clockwork than humans, in historical terms portable timekeeping was basically contemporary with the usual armor technology. The 1e 'great net equipment list' includes a 20 pound water clock costing 1000gp, which seems reasonably portable and 'adventurer affordable' (certainly workable on a ship, wagon, or portable hole), so for at least some versions of D&D portable timekeeping is standard.
A water clock is portable? Like the hour glass it depends on being kept upright (so carrying it in a pack is out), but has the additional problem that you can't jostle it significantly: It will spill or slosh if carried on, say, horseback. And if you've ever carried a pack you know that 20 pounds is a significant chunk of weight to add to your already full pack full of food and supplies.
Back to the main topic though, item recharging may or may not be tied to a celestial event. DM's call. But Clerics getting their power is. They can't do their "prayer and meditation" thing whenever they get around to it. Taken literally, if it's supposed to happen at "sunrise", it happens at sunrise, and if the Cleric misses it for whatever reason, then they missed it and have to wait for it to come around again. But whether it's tied to something like the actual rising (or setting) of the sun is for the DM and player to work out when the character is created. If it's at a particular hour, say 07:17 am (because of its theological significance), and you're someplace where you can't see the sky, or you're in the Arctic and it's overcast, or on the Plane of Shadows (where it's always twilight), then the character really needs to know the time, and that's where my reference to a "portable timepiece" comes into play.
On a somewhat lighter historical note, during the American Civil War watches were valued, but expensive. A professor Thomas Alexander (from Columbia University, if I recall correctly) came up with a battlefield alternative. He developed a chemical mixture that, once applied to a piece of cloth, would slowly change color over time. That way groups of men could coordinate maneuvers with nothing more exotic than a scrap of rag tied around their arm.
Don't tell me that you've never heard of Alexander's rag time band?
https://www.youtube.com/watch?v=Ni6vV9heJhM
