Despite your tags, you asked how common the funny meaning in British English.
Hysterical would normally mean very funny in the more common uses in British English, although it does depend on the context. In common speech, I doubt most people would understand it to mean anything else, despite its history.
Affected by or deriving from wildly uncontrolled emotion
informal Extremely funny
That context is everything, however. If there's any hint of humour, then it's clearly one meaning; if there's the whiff of madness, it's another. Consider:
"I went to see the new Ben Stiller film - it was hysterical"
(with apologies, as this is clearly a ridiculous scenario, as anyone who's ever seen a Ben Stiller film would attest).
vs
"I saw Ben Stiller being restrained - he was hysterical"
vs
"I went to see Ben Stiller - he was hysterical"
The first of these would universally be understood to mean "it was very funny"; the second is clearly suggesting that he was emotionally out of control; the third is ambiguous, as it could mean either. You'd need to decide from the rest of the conversation.
There are historical, sexist connotations - the "women are hysterical" view (as per the womb comment and here) which may change understanding, dependent on the listener. Consider:
"My wife was hysterical"
vs
"My husband is hysterical"
Depending on their prejudices, people could be more likely to understand the former as "emotionally over the top" and the latter as "very funny". Welcome to the modern world.
EDIT
You're still concerned about this topic, so I'll address your questions very specifically - at least, from a UK perspective.
> How common is the usage of the term hysterical meaning “funny” in BrE and AmE?
Very. Most British people would understand hysterical to mean funny as a primary definition, despite what dictionaries say. US preference is hilarious for that use, although that sounds slightly sarcastic to UK ears.
> Does hysterical actually carry a negative connotation as suggested in the above extract, or does it only convey a neutral meaning?
Nothing negative about it (again, BrE). As per the sarcasm comment, it's perhaps even less negative than some other options.
> Is its usage currently on the downtrend?
I see no downtrend in common usage in the UK. Ngram suggests a slight decline in overall use of the word (i.e. whatever the meaning) in both BrE and AmE, but not significantly. Specific to the "funny" use, though, that's certainly not something that's born out in everyday speech in my experience.
Ngram UK
Ngram US
... with a slightly higher incidence in all-English fiction usage.
Best Answer
For a term that has—according to Merriam-Webster's Eleventh Collegiate Dictionary (2003)—been around since 1845 as a noun and since 1880 as an adjective, outsize/outsized didn't make much of a dent in Ngram frequency graphs until around 1920. Over the past century, however, the trajectories of outsize (blue line) and outsized (red line) have been quite interesting:
After holding a substantial edge in frequency of appearance in print from 1920 to about 1980, outsize saw it frequency plummet in the early 1980s, only to stabilize gradually after 1990 and increase modestly in the early 2000s. Meanwhile, outsized rose steadily until 1980, then endured a slight drop in frequency for a couple of years before recovering and rising faster that ever starting in about 1990. It passed outsize in frequency in 1994 and as of 2019 seems to be a bit more than twice as common as outsize in recent Google Books database publications.
According to the Eleventh Collegiate, outsize may function as either a noun or an adjective, whereas outsized functions only as an adjective. It seems reasonable to doubt that noun use of outsize has been very significant over the past sixty years, given the vanishingly small frequency of occurrence of outsizes since about 1960 in the following Ngram chart for outsize (blue line) versus outsized (red line) versus outsizes (green line) for the period 1900–2019:
But even if we suppose that all instances of outsize in print involve use of the word as an adjective, outsize as an adjective is now considerably less common in print than outsized as an adjective.
Even in 2003, Merriam-Webster listed outsize in the preferred position of first spelling in its entry for the adjective:
MW has followed that preference since introducing an entry for outsize/outsized as an adjective in the Seventh Collegiate (1963). Real-world usage, however, may be ignoring the prescribed preference in the Eleventh Collegiate and in other dictionaries, such as the American Heritage Dictionary of the English Language, fifth edition (2010).
Only one usage commentator I consulted took up the question of outsize versus outsized—and his treatment is brief to the point of being peremptory. From Bryan Garner, Garner's Modern American Usage, third edition (2009):
When a variant form becomes twice as frequent in print as the standard form, it may be appropriate to revisit the question of which form is standard and which is a needless variant—or whether the concept of a "needless variant" is useful in appraising the situation.
An argument in support of a form on the basis of historical precedence may exert some emotional pull, but it can't stand up to popular usage in the long term. Thus, for example, a search for hath at Early English Books Online (which covers books published in English no later than the seventeenth century) yields 2,604,169 matches in 47,027 records, whereas a search for has yields 423,473 matches in 18,375 records. So you might have argued in 1700 that hath was standard and has was a needless variant, even though momentum was already on the side of has by 1700. At some point, however, the argument that hath is standard because of its older claim to being standard fails against the overwhelming fact that has eventually became much more popular.
Today, outsize still has a substantial number of adherents (I'm one of them). But on the evidence of the Ngram plots, I am not inclined to agree with Bryan Garner that outsize is standard and outsized is a needless variant.