[Legate of Amon Lanc]

I think hewhoarisesinmight is right about the medieval measure of time. Concerning hobbits - as we know, hobbits had many things which were not that much medieval (umbrellas...), so perhaps they had even mechanical clock? It seems clear however that in the Shire the time was measured from midnight to midnight (or more likely from midnight to noon as AM and from noon to midnight as PM), because it is not stated only in The Hobbit, but also in LotR. So, no only-Hobbit-contradiction here! The most obvious part speaking of counting of time is this:

Originally Posted by RotK, Chapter 1: Minas Tirith

With that Gandalf went out; and as he did so, there came the note of a clear sweet bell ringing in a tower of the citadel. Three strokes it rang, like silver in the air, and ceased: the third hour from the rising of the sun. After a minute Pippin went to the door and down the stair and looked about the street. (...)
"Nine o'clock we'd call it in the Shire," said Pippin aloud to himself.

Evidently here we face the typical time-measure system based on the movements of the Sun. But be this only Númenorean/Gondorian habit or be it used in more parts of M-E, we don't know. Taking into account the Númenorean enterprise and their impact on the whole world, this system of measure might have been adopted by other cultures as well. But maybe not, counting the time from dawn was not such a typical thing. Some cultures in our world, also in ancient times, also measured the time from dusk - the Jews, for example. Now speaking of it, I recall the Elves also did. Indeed:

Originally Posted by LotR Appendix D, The Calendars

A "day" of the sun they called ré and reckoned from sunset to sunset.

So there might have been cultures who have measured the time many different ways, as much as in our world. About the Easterlings or Southrons we can only speculate, for example. And Orcs and Dwarves? I can imagine a Dwarf very well constructing a mechanical clock, since they being stuck somewhere underground would have no other way of keeping track of time (or - of course they would, by water clock for example, but this would be very, let's say, precise and also - dwarfish.) Maybe the Hobbits could have obtained the knowledge of the mechanical clock, as well as the 24h system, from them? (the Dwarves living in Ered Luin) This would explain why such system did not reach the other, even more "civilized" parts like Gondor (or even Arnor at the time of its glory). This would also imply that the Stoors from Anduin would still count time in some other, more "primitive" or better said, "natural" way.

Only speculating, though. We have no evidence of mechanical clock in ME - or have we?
(goes to look deeper)

Wow, a really interesting topic. I really dived into it