For example, ‘speaker’ sounds like ‘speager’ and ‘Stop it’ sounds like ‘stob it’.
Would it have to do with the stop in BE, at least for the first word?
When I pronounce the words with my fake British accent (not cogney, but not quite proper either) "speaker" sounds lie "speek-uh" (apologies -- I don't have the wherewithal to include a proper phonetic spelling) and pronouncing "stop it," I hear "Stawhpit."
In my (inexperienced) opinion, the Queens English has more uniformity in phonemes than American English--the vocalized consonants are vocalized, the unvocalized consonants are not--"Stawh-pit" (where pit is as close to peet as I could get without changing the letter to ee) as opposed to one of the American Northeast metropoli (where everything is fast and furious) "Stobpit" (where the I is clearly a short I). Likewise, "speak-uh" becomes the rushed "speagker." I would also enter the following supposition--in American English of the NE, besides rushing through the sentances, clipping the unvocalized consonants into vocalized consonants to save time, we tend towards laziness in conversation--it is not so much that we don't pronounce K or P, but that they are partially (or wholly) vocalized making them sound like G or B, and easier to pronounce quickly.
Another example of laziness in conversational pronunciation: In Philadelphia, for example--"did you eat?" becomes duhjeeet" where I would almost leave out the "uh" and replace with an apostrophe. The second D sounds like a J, even though original phrase contains no letter J, or G.
As a side note, High German (of Hanover I believe,) has strict pronunciation rules, which make it much easier (for me) to learn the pronunciation of new words. I'm sure many a foreigner wishes English was like that.
It's a "lazy" accent - but it is not unique to the USA. However, it may be that you are only seeing exaggerated / stereotypical accents for non-US English speakers in media you are consuming.
If we take the first example, "Speaker" verses "Speager" - the first two phonemes ('/s/', '/p/') are unvoiced constants, the third ('/i:/') is a long, unvoiced vowel, and the fifth ('/ə/') is an unstressed, unvoiced vowel. The fourth phoneme, the '/k/', is a plosive (unvoiced velar plosive). This takes both more time and effort than using a stop, such as '/g/' (voiced velar stop - as this stop is not followed by a plosive, it is sometimes referred to as a "voiced velar applosive"). Instead of just obstructing the airflow, you have to obstruct it, build pressure, and then release. (In actuality, you'll find that this tends to be an unvoiced velar applosive - neither '/k/', nor '/g/' - which avoids the "effort" of using the larynx / voicebox)
Taking the second example, "stop it" versus "stob it" (or "stobbit"), a similar trend is shown: The unvoiced bilabial stop ('/p/') is sandwiched between two short voiced vowels ('/ɒ/' and '/ɪ/') - instead of interrupting the flow of the word to properly "unvoice" it - i.e. no longer using the larynx - the voiced bilabial stop, '/b/' creeps in. Again, this takes slightly less time and effort than saying it correctly would.
(This also correlates to why accents often "fade" to this style while singing)
The most well-known example of a non-plosive ("applosive") stop is the "Glottal Stop" (ironically sometimes called a "Glottal Plosive"), such as when pronouncing "butter" as "bu'er".