EDIT: It has come to my attention that this is not true apposition, which would be 'one of them, those people.' Even so, my explanation may give you some idea of what you are looking for, and so I will leave it.

'One of them people.'

The two objects, them and people, are in apposition to each other. Each of them is a noun and each serves the same grammatical function; the purpose of each one, however, is to clarify the identity of the other.

In Modern English, it is unusual to find pronouns (them) in apposition; however, in other languages, it is quite normal. When I studied Old Icelandic I often came across such constructions as:

'He Authun went to see them his friends.'

'She his sister went to see him Authun.'

In Modern English you will see apposition in many constructions, usually in titles, but rarely with pronouns:

'King Ethelred was the rightful king.'

'Emperor Julius Caesar was supposedly born via Caesarian section, whence the name.'

'The cook, John, likes his own soup more than we, his customers, do.'

Or, less commonly:

'Look at them stars.'

But never in Modern English:

'They stars are looking down at us.'

A different analysis may simply say that them people is a colloquial variant of those people, and, to some extent, it is; but I thought that you might want a more thorough explanation of what exactly was happening here.

I did not answer all of your questions, but I do hope that this helps.

The other two answers have addressed whether the construct is "grammatical" or not, so I wanted to tackle the other part of the question:

Can we use it in daily speech? Can this usage go beyond a specific dialect and be used in other dialects, regions etc.?

In my experience, within the American South and rural Appalachian dialects it is certainly used in daily speech. There are certain idioms where it's irrespective of region (e.g. "One of them days.") though sometimes it has an ironic flavor to it.

But in some regions (such as the Mid-Atlantic or Northeast America regions), using the construct would definitely raise eyebrows. You might be looked upon as quirky (at best) or uneducated.

I do find it interesting that even in the Northeast region where adults don't speak that way, children still do. I commonly see toddlers say things like: "I want them trucks". Of course they also use it improperly as a subject: "Them trucks are pretty." Or even "He is them friend." Makes me wonder if the origin has something to do with some kind of simplistic mis-application of the objective case.

There seems to be a natural tendency for English speakers in colloquial speech to use the objective case in place of the subjective in stock phrases, particularly for emphasis: "It's me", "That's him", and so forth. Over time this will might lead to a loss of the subjective forms altogether, as has already happened with you (objective form of the now archaic ye). Speakers who use them in the way you describe may just be a bit further along in this process.

etymonline.com says "them" is related to Old Norse theim written with the special letter thorn, the th-sound as in "them". theim was the dative plural of "they". What I miss is a hint at German:

Compare they die, them denen, their deren/derer. And compare one of them einer von denen, meaning one of those people there.

In English them was used as object case and in dialects "them" is also used for those. Compare the title "Them!" of the science fiction film with the giant ants, USA 1954, directed by Gordon Douglas. Actually the little girl that cries Them! means: Those (giant monster ants)!

This is from personal experience:

"Them" takes the place of "those" in plain speech. Plain speech in America has very strong appeal. Nixon attributed most of Truman's charisma to his plain speech.

Bear in mind that using "them" in the place of "those" will confuse ESL speakers nine times out of ten.

The usage of "them" as oppose to "those" for the English language comes from the same bastardization (bastardisation for the UK English) of replacing 'my' with 'me'. It is taken from the fact that over time words in the smaller English dialects have been intertwined with modern English to give some mixture of the two. As you can tell, I think it is abhorrent and should be avoided at all costs.

But how does all this apply to this particular subject? Well, the usage is where the problem lies. Because of the medley of words and languages fused within English, the form and structure of sentences is just as diverse. Many smaller villages, including some of the highly educated ones (Oxford and Cambridge, both of which are the home of their respective dictionaries, to name a couple) use an improper English structure. Why would they replace the simple word 'my' with the incorrect 'me'? E.g. "That's me hat!", "That's me car!", "That's me mam!"

Like with all Linguistics studies, the arguments of the form and origin of this type of statement cannot be defined accurately because of how English has ended up being a hodgepodge of multiple languages all rolled up to help express ourselves. There are plenty of words that are borrowed from other languages. Still, I doubt that the origin of this comes from what would have been known as the New World in those days. During the Colonial era, English was not the primary language of the Americas as the French, the Spanish, the Dutch and Germans, etc, had all arrived to stake a claim. I think that this particular use of the language finds itself being brought over from British Isles, perhaps in the form of lower class cockney from the little or even uneducated masses who struck out to the Americas in hopes of finding a new start to their lives.

As to it's propagation? Hereditary. I believe that its origins come in the form of laziness. Yes, laziness. Nothing overtly complex nor extravagant. In much the same way Americans today say moun'ain by refusing to pronounced the 't' in the word, the effort of circling your lips to push out the 'oh' in 'those' is more effort then the 'eh' in 'them'. Over time this habit has been passed down and is now considered 'normal' (if one can consider it that), but has stayed in the rural areas. Over time, the major cities attracted the best and brightest of the world and as a result the standard of the usage has been kept up.

Of course, this is speculation, but I'll keep an eye out for sources that may or may not support this theory.

The simple and honest answer is no, it is not correct. In England only those who have been poorly educated use the word "them" in this way.

It's definitely incorrect to use "them" instead of "those" in the above examples. As an Irish man living in England it never ceases to amaze me how certain English people cannot even speak their own language correctly, but as Oscar Wilde said "the only good thing the English gave the Irish was their language, and then we taught them how to use it!".

My take on it is that the use of "them" originates from the north of England and also from the East End of London/Essex, and that an inverse snobbery has taken hold wherein it's so common-place to use "them" that a whole generation has grown up thinking that this usage is grammatically correct.