How do you proceed from pronouncing "t" in the regular way to t-glottalization, as found in various English accents?

Short answer: The transition from /t/ to glottal stop does not require intermediate steps.

Explanation:

There are three main factors involved in the production of a consonant: place of articulation, manner of articulation, and voicing. The glottalization of /t/ is essentially a loss of place.

/t/ is a voiceless alveolar stop. These three words in the name represent the three main factors. Voiceless means there is no voiced 'hum' (as in /d/). Alveolar means that the sound is produced on the alveolar ridge (the place). Stop (also called plosive) means that the sound is made by a full closure of the oral cavity followed by a burst of air.

/ʔ/ is the (voiceless) glottal stop. As you can see by the name, 2 out of the 3 parts of articulating the sound are the same as /t/. Glottal sounds are produced in the glottis itself.

When a consonant that is normally (or formally) articulated in one part of the mouth is articulated instead in the glottis, this is a form of lenition known as debuccalization. As you can see, lenition and debuccalization are phenomena that occur independently in a vast number of languages. It is essentially a "weakening" of a consonant.

Since this is simply the loss of one feature, there are no intermediate steps — except that, in cases of a total and permanent loss of place in some context within a language, there is usually (if not always) a period where there is free variation between the lenited form and the full form, until eventually the full form becomes so rare it falls out of use.

In American English, we currently have free variation in the way we pronounce /t/ at the end of many words, like get, hat, astronaut, and so on. In careful pronunciation, a full /t/-sound (closure followed by release) is made. But often the /t/ is articulated but never released, and in many dialects (including my own), the /t/ is realized as [ʔ], the glottal stop. So, I often say [gɛʔ], [hæʔ], [æstrənɔʔ], and so on. This is an example of an intermediate stage of lenition. Perhaps, in a couple hundred years, speakers of my dialect will never pronounce the /t/ in that context.

In American English, word-final stops like /t/ are commonly glottalized, so that "cat" /kæt/ is pronounced [kʰæʔ͡tˢ]. More precisely, it is realized as a co-articulated glottal and alveolar stop.

In many environments, this final stop is unreleased. Acoustically, then, there's little difference between unreleased [ʔ͡t̚] and just a plain glottal stop [ʔ] (it's the /s/-colored release that gives it distinctness), so it's not surprising that the simpler articulation is often used. (The degree this occurs varies considerably by dialect.) The /t/ -> /ʔ/ phenomenon is thus a loss of the alveolar articulation.

A special case is when /t/ is followed by /n/, as in cotton /kɑ.tn/. Plosive + nasal sequences assimilate so that the stop is unreleased; the resulting articulation is alveolarly stopped for the whole duration, during which the nasal passage opens and voicing begins. If the glottis were left open for the whole time, then the pressure from the lungs would produce an uncomfortable nareal plosive when the nasal passage opens; hence it is pronounced glottally stopped. In this case, there isn't even any articulatory difference between /ʔ͡tⁿ/ and /ʔⁿ/.

My days of formally studying linguistics are long gone, so I don't speak with the authority of a professional linguist on this particular one. But I've no reason to assume there's some evolutionary process at work here like the Great Vowel Shift.

People just hear a sound and attempt to reproduce it. Sometimes they don't really care if the phoneme they generate isn't identical to the target, so long as it's close enough to be understood. Even parents eventually give up if their kids won't roll their r's or pronounce their aitches properly.

And - this may be purely subjective - I think the glottal stop is just easier in a lot of vocalisations. So long as your audience don't think you're 'common'.

The Yorkshire accent misses out 'the' almost completely. If you are writing comedy Yorkshire accent you would write "going on t'internet" but it's pronounced with just a very short stop in place of the 't'

I assume this is because the 'the' is completely unnecessary and so Yorkshire people don't see why they should waste words saying it.