The conductor was Sir Edward Downes, and his wife was Lady Downes. The story never uses the expected plural of Downes: Downeses. (The rule is for a noun ending in -s, add -es to make the plural form.) The story seems to avoid the plural, referring to the Downes family.
However, John F. Burns' account contains this passage:
But British news reports about the Downes’ suicides noted one factor that appeared to set the case apart from others involving the Dignitas clinic: Sir Edward appeared not to have been terminally ill. There have been at least three other cases similar to the Downes’, in which a spouse who was not terminally ill chose to die with the other.
Because Burns is referring to both Downeses, it seems that the plural possessive would be Downeses', instead of Downes'. If I were editing the story, I'd probably want to change Downes' to Downeses' in that passage.
The Associated Press Stylebook says to add -es to form the plural of a proper name ending in -s, and says to add an apostrophe to form the possessive, although it doesn't give an example like Downeses. The Chicago Manual of Style does give an example that seems to apply: the Williamses' house. Bryan A. Garner in "Garner's Modern American Usage" and the Economist's Style Guide give an example, too: Joneses'. And, most important to this particular story, The New York Times Manual of Style and Usage gives Joneses' as an example.
But the Guide to Grammar and Writing has this:
When we want the possessive of a pluralized family name, we pluralize first and then simply make the name possessive with the use of an apostrophe. Thus, we might travel in the Smiths' car when we visit the Joneses (members of the Jones family) at the Joneses' home. When the last name ends in a hard "z" sound, we usually don't add an "s" or the "-es" and simply add the apostrophe: "the Chambers' new baby."
So does that apply in this case? It would seem that Joneses and Downeses should be treated the same way. I don't think the "z" rule applies.
Besides, I would probably still use Chamberses' to indicate something that belongs to the Chambers family. This "z" rule stuff seems to lead us down a twisted path of inconsistency.