Meaning of term ‘rapidly labeled RNA’
Explanatory answer

Rapidly-labelled RNA was the RNA labelled by radioactive precursor during a short period (the pulse), terminated by swamping the system with an excess of non-radioactive precursor (the chase). This is known as a pulse–chase, and is described in the methods section of the paper cited in the question and reiterated by @SeRe as an answer. Such a pulse–chase preferentially labels RNA species that have short turnover times (i.e. half-lives of the same order of the pulse), and was thus regarded as a way of studying messenger RNA. 

This strategy and its interpretation was an extrapolation from what was known about mRNA in the regulation of bacterial gene expression, but the descriptive term ‘rapidly-labelled RNA’ was used rather than ‘mRNA’ because there was no direct evidence to support such an assignment in nucleated mammalian cells. Indeed, there were major problems because the rapidly-labelled RNA in the nucleus was of a different size to the RNA that subsequently appeared in the cytoplasm, and, indeed, there was no simple precursor–product relationship between the two. In retrospect this is not surprising given what we now know about the splicing of eukaryotic mRNAs and the fact that most of them have much longer half-lives than bacterial mRNAs. However, at the time there was a school of thought that interpreted these results as evidence against the idea of mRNA in eukaryotes, and one of the purposes of the paper cited was to demonstrate that some of the rapidly-labelled RNA that appeared in the cytoplasm had properties consistent with mRNA.

The historic context

It may be necessary to explain to younger scientists that this work was done before DNA cloning, before RNA sequencing (never mind DNA sequencing), before the discovery of polyadenylation of eukaryotic mRNAs (the ‘handle’ for purifying them), in short, at a time when no specific mRNA had yet been identified or isolated (see, e.g., the Wikipedia article on The History of RNA Biology). As documented in Mathew Cobb’s excellent article ‘Who discovered mRNA?’, mRNA wasn’t really discovered but was an idea to account for the rapid response of the lac system (observed in genetic experiments) in terms of a previous unexplained labelled RNA with a DNA-like base composition observed associated with more abundant stable RNA (ribosomes) when bacteriophage infected E. coli. Subsequent work, presented in papers by Brenner et al. and Gros et al. in the same issue of Nature in 1961, in effect documented this RNA more precisely in phage-infected cells and showed that it differed from ribosomal RNA. The importance of the work of Mathhaei and Nirenberg in establishing cell-free systems for translating polyU and subsequently other artificial polynucleotides cannot be underestimated. However, it is important to recognize that the genetic code was not deciphered using genuine mRNA.

It was natural to assume the mRNA model extended to eukaryotic cells, but at that time there was no convenient system with an abundant mRNA like that of bacteriophage and so the presumed differentiating characteristic of short half-life was used, as seen in a plethora of papers on ‘rapidly-labelled RNA’. Many of these were opportunistic, claiming to document anticipated hormonal or other effects on eukaryotic gene expression. Consequently the demonstration that much of the rapidly-labelled RNA in the nucleus never reached the cytoplasmic ribosomal location had the effect of discrediting not only these papers, but the very idea that mRNA existed in eukaryotes. This may seem rather absurd today, but the foremost ‘eukaryotic mRNA denier’ (as he might be termed in contemporary parlance) was the eminent Oxford Professor of Pathology, Henry Harris FRS, who developed man–mouse heterokaryons and set out his ideas in a published monograph entitled ‘Nucleus and Cytoplasm’ (a copy of which I once possessed). So it is relevant that Warner et al. (1966) refer specifically to Harris in the discussion section of their paper, which describes work that addresses the issues Harris had raised.

If there is a moral to this story (and it is always easier to draw morals after the event) it is perhaps that Harris was happy to confusticate lack of proof with disproof in an area that was peripheral to his main concerns, and that the groups in New York — and Jonathan Warner, especially, having previously described polysomes in Alexander Rich’s lab — were prepared to accept the fact that the situation was more complicated than originally thought, but were not prepared to throw out the baby with the bath water.