Unfortunately this is a very difficult question to answer, partly because of Tolkien's framing device.
The in-universe origin for the Tolkien Legendarium is that Bilbo and Frodo wrote down their stories (The Hobbit and The Lord of the Rings, plus Bilbo's translation of Elvish legends that would become The Silmarillion) in the in-universe Common Speech. Tolkien later discovers these ancient texts, and laboriously translates them into English.
This unfortunately means that it's hard to take any mention of timekeeping at face-value, because a reference like the following one in Fellowship of the Ring:
'Where am I, and what is the time?' [Frodo] said aloud to the ceiling. 'In the House of Elrond, and it is ten o'clock in the morning.' said [Gandalf].
Fellowship of the Ring Book II Chapter 1: "Many Meetings"
necessarily comes with the caveat that we don't know to what extent Tolkien is exercising some discretion as a translator1
Having said that, what do we know?
Are there clocks in Middle-earth?
Yes. There's a mention of a clock in Fellowship of the Ring, when Bilbo leaves the Ring at the very beginning:
Bilbo took out the envelope, but just as he was about to set it by the clock, his hand jerked back, and the packet fell on the floor.
Fellowship of the Ring Book I Chapter 1: "A Long-Expected Party"
This clock is referenced several more times in The Hobbit. Even if we allow this to be the translator taking liberties, it's very probable that Bilbo had some kind of personal time-keeping device. Tolkien does us one better, with a 1937 illustration of Bilbo's front hall featuring at least one wall-mounted clock2:
How granular is time subdivided?
Looking at the sketch above, there do appear to be two individual hands indicating a concept related to hours and minutes.
We can further support this with references to various times made by the hobbits. Pippin, for example, wants his breakfast at 9:30:
'What's beautiful about it?' said Pippin, peering over the edge of his blanket with one eye. 'Sam! Get breakfast ready for half-past nine! Have you got the bath-water hot!'
Fellowship of the Ring Book I Chapter 3: "Three is Company"
Pippin also makes reference to minutes in The Two Towers, telling the story of his first encounter with Gandalf the White:
'"Treebeard," said Gandalf. "I need your help. You have done much, but I need more. I have about ten thousand Orcs to manage."
'Then those two went off and had a council together in some corner. It must have seemed very hasty to Treebeard, for Gandalf was in a tremendous hurry, and was already talking at a great pace, before they passed out of hearing. They were only away a matter of minutes, perhaps a quarter of an hour. Then Gandalf came back to us, and he seemed relieved, almost merry. He did say he was glad to see us, then.
The Two Towers Book III Chapter 9: "Flotsam and Jetsam"
There is only one reference to seconds, as when Merry threatens Bill Ferny to get into Bree:
'Bill Ferny,' said Merry, 'if you don’t open that gate in ten seconds, you'll regret it. I shall set steel to you, if you don't obey. And when you have opened the gates you will go through them and never return. You are a ruffian and a highway-robber.'
Return of the King Book VI Chapter 8: "The Scouring of the Shire"
Regarding the exact composition of days, hours, and minutes: we can't be sure. There's some information about calendar deficits in Appendix D that could potentially be used to calculate how many seconds in a minute and how many minutes in an hour, but I don't have a head for those sorts of calculations.
What script would the numbers be written in?
Let me first point out that clocks of this sort seem to be unique to hobbits3, so this part comes down to what writing system hobbits use.
This is never stated explicitly, but it's most likely the Tengwar characters; in Appendix E, Tolkien says that all cultures in the Third Age used writing systems borrows from the Eldar, and that the Tengwar was used pretty much everywhere the Common Speech was used:
The scripts and letters used in the Third Age were all ultimately of Eldarin origin, and already at that time of great antiquity.
The later letters, the Tengwar of Fëanor, were largely a new invention, though they owed something to the letters of Rúmil. They were brought to Middle-earth by the exiled Noldor, and so became known to the Edain an the Númenoreans. In the Third Age their use had spread over much the same area as that in which the Common Speech was known.
Return of the King Appendix E "Writing and Spelling" II: "Writing"
Would the clocks be in base 10 or base 12?
This is a surprisingly fascinating question.
In Appendix D, Tolkien suggests that the Elves prefer a duodecimal (base 12) number system:
It seems clear that the Eldar in Middle-earth, who had, as Samwise remarked, more time at their disposal, reckoned in long periods, and the Quenya word yén, often translated 'year' (I, p. 496), really means 144 of our years. The Eldar preferred to reckon in sixes and twelves as far as possible.
Return of the King Appendix D "The Calendars"
However the Númenóreans weren't keen on that calendar system, and modified it to one that would be more familiar to us, and a note in Unfinished Tales indicates that they preferred a decimal system:
Measures of distance are converted as nearly as possible into modern terms. "League" is used because it was the longest measurement of distance: in Númenórean reckoning (which was decimal) five thousand rangar (full paces) made a lár, which was very nearly three of our miles.
Unfinished Tales Part III: The Third Age Chapter 1: "The Disaster of the Gladden Fields" Appendix: "Númenórean linear measures"
From Appendix D we learn that the hobbits modified their calendar from the Númenóreans:
The Hobbits were conservative and continued to use a form of the Kings' Reckoning adapted to fit their own customs.
Return of the King Appendix D "The Calendars"
Thus it's most likely that it was base 10.
1 Darth Melkor (user8719) talks about this issue a bit more in his answer here
2 The other object is possibly another clock, as suggested by Michael Martinez, but DJClayworth in comments suggests that it's more likely a barometer
3 Elves and Men seem to use a system of bells for keeping time, and it always seems to either mark events rather than time (as with the Elves) or reckon hours relative to the sun (as in Minas Tirith and among the Rohirrim). Unfortunately we don't know what the Dwarves use for timekeeping.