Psychologists followed up on this by attempting to test this hypothesis with a number of language pairs (for an overview, see Baddeley (1997). Disclaimer: I'm not a psycholinguist, and the following references are certainly not exhaustive). The best-tested and most consistent result concerns Chinese. Mandarin and Cantonese numerals take shorter to say than English ones, and a number of psychologists have accordingly confirmed that Chinese speakers can remember longer numerals than English speakers (Stigler, Lee, & Stevenson (1986), Hoosain & Salili (1987)), even at 4 years old Chen and Stevenson (1988)), and that this applies even when bilinguals are tested across their two languages (Hoosain 1979). It goes further than that, in fact: Chincotta & Underwood (1997) find that, out of Cantonese, English, Greek, Finnish, Swedish, and Spanish, only Cantonese speakers remember significantly more digits than speakers of other languages - and that this difference disappeared if the subjects were prevented from rehearsing the numbers auditorily by being asked to keep repeating "la-la" while being tested, proving its linguistic nature. The difference ranges around 2 digits, with the exact figure depending on the experiment.
Data for other languages is less clearcut. Welsh numerals take longer to say in isolation than English ones, and Ellis & Hennelly (1986) accordingly found that English-Welsh bilinguals can on average remember longer numerals in English than Welsh. Naveh-Benjamin & Ayres (1986) simultaneously tested the hypothesis for university students in Israel speaking English, Spanish, Arabic, and Hebrew natively (but excluding the digits "seven" and "zero"). They found that the average number of digits recalled was highest in English (7.21), followed by Hebrew (6.51), then Spanish (6.37), and lowest in Arabic (5.77); the ordering by average number of syllables per digit, or by average time taken to read a digit, was English, Spanish, Hebrew, Arabic. However, the difference in number of digits recalled was smaller than predicted by the time taken to read a digit in each language, suggesting that other factors were also relevant.
A proviso is necessary: some recent work, without disputing the differences observed, has made a strong case that they relate not simply to length ( Lovatt, Avons, & Masterson 2000), but crucially to phonological factors (Service 2010, Lethbridge, Hinton & Nimmo 2002). This has been argued for Welsh numerals vs. English ones by Murray & Jones (2002), who find that Welsh digits take longer to say in isolation but actually take less time to say in connnected speech than English ones, and that changes of place of articulation at word boundaries negatively affect memory.
The research is curiously selective in terms of languages examined, and many of the experiments don't control for all possible confounding factors, such as diglossia and social status in the case of Welsh or Arabic. Nevertheless, it does at least seem well-established that speaking Chinese gives a short-term digit memory advantage over speaking major European or Semitic languages. So, if for some reason you regularly need to remember long numerals, and your preferred language doesn't happen to be Chinese, how do you compensate for this handicap?
There are two obvious ways to get around this (assuming you care enough about remembering numerals to want to, which depends very much on your tastes and circumstances). One is to remember the number visually (as a sequence of written digits) or even kinesthetically (as a sequence of typing actions), in which case this particular constraint no longer applies (cf. eg Olsthoorn, Andriga, & Hulstijn 2012). This only helps, however, if you remember numerals better visually or kinesthetically than auditorily, and my impression is that most people don't.
A probably more helpful alternative is to establish a code that lets you turn long numerals into much shorter words by identifying digits with single letters or single phonemes. This solution has a very long history in Arabic and Hebrew, in which each letter of the alphabet can be used to represent a digit: 'a is 1, b is 2, etc. (the first 9 digits are units, the second 10 are tens, and the rest are hundreds). Since short vowels are not letters, the resulting word can be given whatever vowels the user sees fit to give it. A common game of later poets using the Arabic script was to encode the date of their poem within the poem as a chronogram; more practically, Moroccan schoolchildren used to memorise the multiplication tables as a series of meaningless words formed by this encoding (Meakin 1905). Chronograms have been formed using Roman numerals, but for memorisation, at least, they are rather ill-adapted to such a system - think how much padding would be required to turn a number like MDCCCLXXXIII into words.
However, the spread of Hebrew studies in Western Europe following the Renaissance, and the increasing importance of memorising statistics there, encouraged European mnemonists to look for ways of emulating this encoding without having to learn a Semitic language. Doing so at a time when place notation was widely used, they introduced a crucial improvement: each consonant represented a digit in a place notation system, rather than a number in an additive notation system. After various cumulative efforts at improvement, this culminated in the early 19th century with the so-called Major system: 0=s/z, 1=t/d, 2=n, 3=m, 4=r, 5=l, 6=š/ž/č/j, 7=k/g, 8=f/v, 9=p/b, with vowels, semivowels, and laryngeals ignored. To remember 94801 (LACITO's zip code), for example, one would turn it into "professed". This system apparently remains in use among professional mnemonists to this day, despite being virtually unknown to wider society.
Perhaps this is why linguists haven't paid more attention to the word-length effect in the context of the Whorfian debate: it's a clear-cut effect of language on cognition, but not a very profound one, in that it should be fixable by some very simple hacks (or even just by borrowing some one else's numerals). But I'm not aware of any experimental work testing the effect of this particular hack on digit recall...