Is there documentation on how long it takes before publications are indexed by Scopus and Web of Science?
This is a really interesting question!!

According to Clairvate Analytics WebofScience/WebofKnowledge), updating of the WOS collection occurs daily on Monday through Friday. This revelation, paired with @Andrew 's comment about Scopus, suggests that the lag time exists with the publisher and not the database compilers (i.e., how long before the publisher submits (or makes available) the DOI and metadata). Here is an interesting op-ed on these lag times.

It is also important to note that articles which are published in non-indexed journal will not appear in most collections/databases, however, GoogleScholar will may pick up more grey literature (e.g., government articles, theses), and potentially some of the non-indexed journals. This article suggests provides 'optimal' updating frequency compared to GS and WOS in their field.

If the article is available online as either an early print or post-print, then it should have a DOI (I assume), and therefore should be searchable via Google. It follows that Google Scholar should pick up articles faster than human-curated databases (Scopus, WOS).

In a brief search (using GoogleScholar), I came across a preprint by Haustein et al. When is an article actually published? An analysis of online
availability, publication, and indexation dates.. The authors found that in some fields lag time has decreased but has increased (!) in others: 


  Various studies have analyzed publication delays and found differences
  between scientific fields, journals, and publishers (e.g., Abt, 1992;
  Amat, 2008; Björk &amp; Solomon, 2013; Das &amp; Das, 2006; Diospatonyi,
  Horvai, &amp; Braun, 2001; Dong et al., 2006). Since long delays interfere
  with priority claims and slow down scientific discourse, publication
  speed plays an important role for authors and scholarly communication
  (Rowlands &amp; Nicholas, 2006; Schauder, 1994; Tenopir &amp; King, 2000).
  Short publication delays can therefore be considered as a quality
  indicator reflecting the up-to-dateness of scientific journals
  (Haustein, 2012). Publishers have begun to reduce delays by making
  so-called early view, in press, ahead of print or online first
  versions of accepted papers available before they appear in an (print)
  issue. It has been shown for food research journals that online ahead
  of print publication has reduced publication delay by 29% (Amat,
  2008), while Das and Das (2006) reported for 127 journals in 2005
  average lags of three months between online and print issues
  publications with particular differences between publishers. Tort,
  Targino, and Amaral (2012) showed that this lag increased
  significantly over time for six neuroscience journals. Online dates
  are now being recorded in bibliometric databases like Scopus, which
  impacts bibliometric analyses (Gorraiz, Gumpenberger, &amp; Schlögl, 2014;
  Heneberg, 2013). 


Various distributions are made available in Table 2 of the aforementioned paper.

An aside: it may be worth checking out some of the works citing Haustein et al.