I understand you read 2000 aloud as twenty hundred hours and 0000 as zero hours.
How then do you read 0001 and 0010?
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Sometimes, instead of zero, oh is used, although this is a more colloquial form. From what I have read, whether or not the "hours" part is pronounced depends on the branch of military; usually it is understood to be implicit and thus left out.
This is an expanded and edited version of my original answer, with additional material included primarily to address various comments below my original answer suggesting that:
The 24-hour clock is the convention of time keeping in which the day runs from midnight to midnight and is divided into 24 hours, indicated by the hours passed since midnight, from 0 to 23. This system is the most commonly used time notation in the world today, and is the international standard (ISO 8601) notation for time of day. ... It is popularly referred to as military time or astronomical time in the United States, Canada, and a handful of other countries where the 12-hour clock is still dominant.
ISO 8601 uses the 24-hour clock system. The basic format is [hh][mm][ss] and the extended format is [hh]:[mm]:[ss].
- [hh] refers to a zero-padded hour between 00 and 24 (where 24 is only used to notate midnight at the end of a calendar day).
- [mm] refers to a zero-padded minute between 00 and 59.
- [ss] refers to a zero-padded second between 00 and 60 (where 60 is only used to notate an added leap second).
So a time might appear as either "134730" in the basic format or "13:47:30" in the extended format.
It is also acceptable to omit lower order time elements for reduced accuracy: [hh]:[mm], [hh][mm] and [hh] are all used.
Although the 24-hour system of writing time is often "popularly referred to as "military time in the United States, Canada, and a handful of other countries where the 12-hour clock is still dominant" 4, in much of the rest of the world it is now the normal format for writing times unambiguously without using "a.m." and "p.m.", especially in schedules, timetables, and many other documents. (In fact, personally, I now get confused when looking at websites that do not display the time in 24-hour format.)
The previous answer gives a link to a website for "Tell-Military-Time", and also refers to possible differences between different of the military. It would appear that that site actually describes usage within the US Military. Personally, I would be hesitant is assuming that any usage described there corresponds to common usage within other English-speaking countries worldwide. That would be especially so, because within the military, they are speaking with others trained in using the same terminology, whereas in the 'outside world' you may be speaking to someone not familiar with particular terminology.
As indicated in the extract quoted previously, the ISO standard permits the presence or absence of separators between the digits representing the hours, minutes and seconds. But it does specify that, when a separator is used, it should be a colon.
In the past, some European countries used the dot on the line as a separator, but most national standards on time notation have since then been changed to the international standard colon. In some contexts (e.g., U.S. military, some computer protocols), no separator is used (e.g., 2359). 5
See the following examples:
Some airlines use a colon separator [Image source]:
Several UK railway timetables use no separator symbol, but use a small space [Image source]:
Several UK bus companies use no separator at all [Image source]:
[I have found many other examples of each format, but will not illustrate or list them here.]
Reverting now to OP's basic question of how to express 24-hour time in speech, military protocols may specify exact formulae, but in civilian usage:
Personally, I would not refer to:
for fear that it may be misunderstood as "one hour" or "ten hours" after midnight. I would probably say:
one minute past midnight
ten minutes past midnight
but, depending who I was talking to, I might use any of the following:
midnight oh one
zero hours one minute
twelve oh one †
one minute past twelve †
† if it were already clear that we were referring to midnight and not noon.
Or any other similar expression that came to mind.
In summary, my answer to the questioner would be that (outside the military and other officialdom) there is no formal set way of expressing times in the 24-hour clock, and I would suggest expressing it in any suitable unambiguous manner that would be understood by the person to whom you are talking, even if that means 'translating' the time into a traditional or 12-hour format.