I think it's not so much an Americanism as prescriptive grammar.

First, Americanism is not mentioned in the relevant sections of either A Comprehensive Grammar of the English Language (Quirk, et al.) or The Cambridge Grammar of the English Language (Rodney Huddleston and Geoffrey K. Pullum).

A Comprehensive Grammar of the English Language

13.39 Restrictions·on correlatives
According to a didactic tradition, the use of correlative coordinators is unacceptable when there are three or more conjoins:

?We are both willing, able, and ready to carry out the survey. 1
?Either the Minister, or the Under-secretary, or the Permanent Secretary will attend the meeting. 2
?Tompkins has neither the personality, the energy, nor the experience to win this election. 3
...
Although commonly stigmatized, multiple correlatives such as [1-3] can add clarity to constructions whose complexity might otherwise cause confusion. For this reason, such constructions are sometimes used even in careful written English, eg in the rubric of an examination paper:
Candidates are required to answer EITHER Question 1 OR Question 2 OR Questions 3 and 4.

The Cambridge Grammar of the English Language

Neither as marker of coordination
As a marker of coordination, neither is usually paired correlatively with nor. It can occur (like either) in multiple as well as the more usual binary coordination, and (like both) it cannot occur initially in a coordination of main clauses:

[48] i She found it [neither surprising nor alarming]. [binary]
ii He was [neither kind, handsome, nor rich]. [multiple]
iii *Neither did he oversleep nor was his bus late. [main clause coordination]

Being descriptive grammars, both CGELs (written by British linguists) endorse the multiple coordination.

Garner's Modern English Usage, Fifth Edition (a prescriptive usage manual written by an American legal scholar and lexicographer), on the other hand, does not endorse the multiple coordination, although it acknowledges there are attested examples of the usage among well-educated people:

B. Number of Elements. These correlative conjunctions best frame only two elements, not more. Though it’s possible to find both modern and historical examples of neither . . . nor with more than two elements, these are unfastidious constructions. When three or more are involved, it’s better not to say *They considered neither x, y, nor z. Instead, say They didn’t consider x, y, or z. Or it’s permissible to use a second nor emphatically in framing three elements: They considered neither x, nor y, nor z. Cf. either (b).
Language-Change Index neither . . . nor with more than two elements: Stage 3

Stage 3: The form becomes commonplace even among many well-educated people, but it’s still avoided in careful usage. Examples include *adopted mother for adoptive mother; *gladiolas for gladioli or gladioluses (or simply glads); idyllic for ideal; *miniscule for the correct spelling minuscule; the supposed contraction *’til for the good old word till (as in We’ll be here till noon); peruse used to mean “scan hastily” rather than “read carefully”; and using a nominative pronoun in compound objects such as *between you and I rather than between you and me.