Mom(m[a/y])/mum(my)/mama and mother are completely different words and the former is not an abbreviation or contraction of the latter.
Compare the other parent: the short, informal words for father are da(d(dy)) and papa/poppa (in different dialects/times). Here it’s more obvious that the ‘full’ word and the ‘short’ words are completely unrelated.
‘Short’ words for parents
The ‘short’ words are very common cross-linguistically: the earliest babblings of babies just beginning to form something that can be taken as linguistic content are selfishly interpreted by the parents as the infant attempting to refer to them, and the parents themselves then start using ‘words’ that are basically imitations of baby babble when referring to themselves with the baby.
Baby babble prototypically consists of repeatedly alternating between the extreme positions of the vocal tract: completely closed (mostly with the lips or the front of the tongue, which is the first part babies learn to fine-control actively) and completely open. Voicing distinctions are usually not mastered yet at this stage, so everything is just voiced. In other words, babies most commonly babble “ba-ba-ba-ba” and “da-da-da-da”. As the parental ‘words’ are really just imitations of this, they too often end up being reduplicative (consisting of the same syllable repeated twice, or at least of two very similar syllables) and having the vowel /a/. For some reason, a remarkably high number of languages in the world settle for the consonant /m/ in the word for ‘mother’ and either a bilabial stop /b p/ or a dental stop /d t/ in the word for father. So very common pairs are mama ~ baba/papa or mama ~ dada/tata.
These words are often not ‘real’ words in the minds of speakers: they tend to behave more like interjections and onomatopoietic words than true content words:
- They are often impervious to otherwise regular sound changes in a language (compare the Germanic languages that all have some variety of papa for ‘dad’ even though /p/ changed to /f/ in Common Germanic, so it ought to be *fafa)
- They often fail to decline the way other nouns do (compare German, where die Mama does not inflect for case at all—it only changes from the singular Mama to the plural Mamas).
This isn’t always the case, though—there are exceptions to both:
- Old Japanese had *papa ‘mum’ and *titi ‘dad’, and both have been allowed to follow the regular phonetic developments of [p] => [h] and [ti] => [tɕi] (written chi) to yield Modern Japanese haha and chichi. (At least almost regular—an original /p/ between two /a/’s should really have yielded a /w/, so ‘mum’ ought to have been hawa if it had been completely regular. Note that Japanese is also not quite in the group of languages described above: while the maternal baby word does have a bilabial, it’s not the nasal /m/ but the plosive /p/, which is more common in paternal baby words than maternal ones.)
- Icelandic mamma and pabbi both decline fully as weak nouns, which means they distinguish in the singular a nominative (mamma/pabbi) and an oblique (mömmu/pabba), and in the plural all four cases: nominative (mömmur/pabbar), accusative (mömmur/pabba), genitive (mamma/pabba), and dative (mömmum/pöbbum).
‘Real’ words for parents
The word mother, on the other hand, is a ‘real’ word—a content word. It is inherited all the way down from the Indo-European ancestral proto-language (where it was *méh2-ter, phonetically probably [ˈmaχter] or something like that), and it has followed the general sound changes that have happened on its way from Proto-Indo-European, through Common Germanic, West Germanic, North Sea Germanic, Anglo-Frisian, and into Anglic and the language we now know as English. The same goes for father.
So why is it mum in the UK and mom in the US? Well … nobody knows. That’s just dialectal variation. The etymonline entry on mamma has the following:
1570s […] Its late appearance in English is curious, but Middle English had mome (mid-13c.) ‘an aunt; an old woman’, also an affectionate term of address for an older woman. […] In terms of recorded usage of related words in English, mama is from 1707, mum is from 1823, mummy in this sense from 1839, mommy 1844, momma 1852, and mom 1867.
So as you can see, there are quite a lot of varying forms of this term, which is quite natural for a pseudo-onomatopoietic word (consider the word used to make someone be quiet: shush, shh, hush, etc.).
As tchrist points out in his comment, you should also note that while mum and mother are spelt differently, they have the same vowel /ʌ/ in pronunciation; conversely, mom has the same orthographic vowel as mother, but a different vowel /ɑː/ in pronunciation.