Yes the law would not have been needed to accommodate the Lawrence family had they heeded the advice they were given warning them against launching a private prosecution. In 1993/4 the CPS had twice declined to proceed as they suggested that, at that time, insufficient evidence existed to support a conviction. The Lawrence family persisted with a private prosecution against Neil Acourt, Luke Knight and Gary Dobson. The three were duly acquitted in 1996 vindicating the CPS stance and (without the subsequent change in the law) putting paid to the three suspects being convicted.
The MacPherson report into the murder and police handling of the case reported in 1999 and among its recommendations (having branded the Metropolitan Police as “institutionally racist" as well as coming up with an absolutely preposterous definition of a “racist incident“) was that the double jeopardy law be re-examined.The rest is history and the double jeopardy law was subsequently abandoned, albeit in the very limited circumstances that Jake describes.