When did the meaning of shi 是 shift from »this« (此) to being a copula (»be«)? I know this question is somewhat unclear since the word shi 是 never only meant this, but carries (and has done so for a long time) several meanings.
The Gu Hanyu da cidian (via Pleco) lists some of them (not a direct quote; the numbers denote the century of the sources that the example sentences were taken from):


  正確(5)；訂正(3)；以爲是(-4)；表示肯定或加強肯定之詞(-3) etc.


But very generally speaking, shi was used in the language of the Lunyu (which I call here Classical Chinese) to denote »this« rather than any of its other meanings.

But when did shi start to mainly have the meaning as a copula that it has today, to be, as in »She is a student«, »Those are books«?
A quote from the 2017 paper "Some New Insight into the Historical Source of the Chinese Copula Shì/*deʔ 是 ":


  Wang (1937) claims that there was no genuine copula in the Chinese language until the emergence of the copula shì / *deʔ 是 around the 5th century — a dating he later modified to the 1st century (Wang 1958) — evolving from the demonstrative shì. 


That is the general consensus, and this idea has been established for the best part of 80 years.

However, there is the statement that:


  [S]ome later studies, especially those based on excavated materials (Qiu 1979; Tang 1991), argue that the copular use of shì starts from the Warring States period (475–221 BCE). 


Li and Thompson (1977) is the classic source for citing this well-trodden pathway cross-linguistically:


  TOPIC - COMMENT > SUBJECT - PREDICATE
  
  [NP1] - [DEM] [NP2] > [NP1 COPULA NP2]


The 2017 paper quoted says that 是 was always somewhat atypical as a demonstrative, always having had anaphoric reference.