This is an interesting question! I can’t find any documentation specifically on the issue, so I don’t have a conclusive answer, but here anyway is a lot of relevant information, and a bit of further speculation.
The OED lists various pronunciations:
Brit. /mam/, /mɑːm/, U.S. /mæm/, /mɑm/; (unstressed) Brit. /məm/, U.S. /əm/, /mˌ/.
It also sheds some light on the recent historical usage:
The γ, δ, and ε forms [respectively mem, mim; mum, mom; and ’m] represent pronunciations formerly common in British regional usage and in the speech of domestic staff and others of similar status; such forms and pronunciations are also well attested in U.S. regional use, especially in yes ma’am (see yessum adv.) and no ma’am, and as the second element in school-marm n. Compare marm n.
Buckingham Palace protocol (c1990) directed that ‘the Queen should be addressed as “Ma’am” (to rhyme with jam).’
It also includes, later:
In 1936, R. W. Chapman ( S.P.E. Tract ii. 241) observed that ‘Except to royal persons, the contraction (whether mahm or măm) seems to be going out.’
This is all informative, but also a bit confusing. The main thing that’s clear is that the variety of pronunciations goes back quite a long way.
The second thing is that in modern BrE, the usage of ma’am is so restricted that it’s very hard to disentangle “what people now use” and “what Buckingham Palace asks for”. It’s a prescriptivist’s dream!
The third is that the OED itself seems a bit confused: the very pronunciation that the Palace asks for, /mæm/, is one which the OED lists only as U.S. usage.
The big question this leaves unresolved is: is it only recently that the Palace has asked for “jam”, or was this already officially preferred in the past?
Another factor which might be involved is how in many words, the vowel /a/ (of farm) is partly shifting to /æ/ (of ham) in prestige BrE accents, and the vowel of jam/*ham* in these accents has changed. In eg the early 20th century, in upper-class British accents, the vowel in ham, jam etc. was more raised and fronted than today, somewhat closer to hem. Conversely, words like glass, class were uniformly pronounced with the long ah vowel, /a/. In many lower-class accents, ham was much like today, with /æ/, and grass, etc. were also pronounced with this vowel.
Since then, the stigma of perceived lower-class and regional accents has decreased, whereas a stigma of being ‘too posh’ has become more widespread; so the raising/fronting of ham is now very rare, and the pronunciation of grass with the ham vowel /æ/ is somewhat more common among RP speakers.
Given that this shift involves many class issues, and the alternation of /a/ and /æ/, I suspect it may have influenced the pronunciation of ma’am somewhat. I’m pretty sure that the shift in the vowel of ham is relevant, given the older spellings of ma’am as mem, mim listed in the OED. The relevance of the change in grass is much more speculative.
Best Answer
As far as I know, pronouncing "subsequent" with stress on the second-to-last syllable has never been generally considered correct. Merriam-Webster, commonly considered a relatively "permissive" dictionary in terms of pronunciation, does not list it, and the old classic of prescriptive pronunciation, Walker's Critical Pronouncing Dictionary (which can be useful as a resource for archaic pronunciations, since the linked edition is from 1791) gives stress on the third-to-last syllable without any mention of other variants. The pronunciation with the accent on the third-to-last syllable is also the only one given by Macmillan, Oxford Dictionaries, the Cambridge Dictionary, the American Heritage Dictionary, and the Collins dictionary.
English stress placement is complex, and I don't know of any argument for considering /sʌbˈsiːkwənt/ "incorrect" that is more compelling than the simple fact that most people don't use it. However, because I find it interesting, I'll attempt to describe the "rules" for stress placement in words of this type. I'm just an interested amateur, so some of this may not be fully correct.
Antepenult stress is, in general, a common pattern for English tri- and polysyllables. Some tri- or polysyllabic words ending in -ent or -ant are always or sometimes stressed on the second-to-last (penult) syllable, but typically this occurs only for words that meet one or more of certain phonological, etymological and morphological criteria. "Subsequent" only meets the morphological criterion, which is somewhat shaky.
Phonological motivation for penult stress: Penult stress occurs most regularly in words of this type when there is a certain kind of consonant cluster after the stressed vowel (as in consistent). But "qu" does not typically count as a stress-attracting consonant cluster (compare consequent and eloquent). Even so, the fact that /kw/ is a consonant cluster might slightly facilitate a variant pronunciation with stress on the second syllable (for comparison, even though "gr" is not typically a stress-attracting consonant cluster, certain speakers are known to pronounce the word "integral" as /ɪnˈtɛgrəl/ or /ɪnˈtiːgrəl/, with penult stress.)
Etymological motivation for penult stress: Less regularly, penult stress can occur when the vowel was long or a diphthong in Latin (as in antecedent, precedent, decedent; however, precedent has an alternative pronunciation with stress on the third-to-last syllable). This reason for penult stress seems somewhat artificial (see the Sargeaunt quote below) and in fact, penult stress is not used in all -ent/-ant words that had a long vowel in Latin. In any case, the second-to-last "e" of subsequent was not long in Latin (it comes from Latin sŭbsĕquēns).
Morphological motivation for penult stress: Sometimes, penult stress seems to occur just because the third-to-last syllable is (part of) a prefix, as in supernatant /ˌsuːpərˈneɪtənt/, from Latin sŭpĕrnătāns = sŭpĕr-nătāns, and covalent, from co- + valent from Latin vălēns.
This might be why the person you heard said /sʌbˈsiːkwənt/: it may be meant to represent the structure of the word as "sub-sequent". Sometimes people consciously choose to use unconventional stress to draw attention to the structure of a word: I had a science teacher in high school who liked to pronounce "hydrolysis" as /haɪdroʊˈlaɪsɪs/ in class to bring our attention to the fact that it meant "splitting by water".
However, it is possible and common for an antepenult syllable that was a prefix in Latin to receive the primary stress/accent in an -ant/-ent adjective—as shown by words like deferent, different, abstinent, competent, innocent, indolent, insolent, applicant—and this is arguably the more regular stress pattern (and the one that the most common pronunciation of subsequent follows).
Some useful comments by rjpond point out that some dictionaries, such as the Oxford English Dictionary and Webster's Third, do record a pronunciation of the related noun "subsequence" with the accent on the first syllable, but an unreduced/secondarily stressed vowel /iː/ in the second syllable.
Here is what John Sargeaunt had to say about the pronunciation of words of this type in The Pronunciation of English Words Derived from the Latin (1920):