To the best of my knowledge, the paper cited by the BBC is "Vanilloid receptor agonists and antagonists are mitochondrial inhibitors: how vanilloids cause non-vanilloid receptor mediated cell death" (PMID: 17214968 DOI: 10.1016/j.bbrc.2006.12.179), which lists Timothy E Bates as an author (the last one, which means he was the primary investigator -- h/t Mad Scientist for correcting my mistake on that), coming from the University of Nottingham School of Biomedical Sciences, and published "2007 Mar 2" (online 2007 Jan 2). The abstract says
These data support the hypothesis that (E)-capsaicin [...] are all
mitochondrial inhibitors, able to activate apoptosis and/or necrosis
via non-receptor mediated mechanisms, and also support the use of
TRPV1 ligands as anti-cancer agents.
As to why this isn't widely used (yet?), that BBC article you linked to says
However, Josephine Querido, cancer information officer at Cancer
Research UK, said: "This research does not suggest that eating vast
quantities of chilli pepper will help prevent or treat cancer.
"The experiments showed that pepper extracts killed cancer cells grown
in the laboratory, but these have not yet been tested to see if they
are safe and effective in humans."
Cancer Research UK recommends reducing the risk of cancer by eating a
healthy, balanced diet, with plenty of vegetables and fruit.
And, of course, things that kill cancer cells in a petri dish aren't necessarily useful as cures.
I searched Google quickly for "capsaicin cancer," and found some more recent articles about it, including "Anticancer Properties of Capsaicin Against Human Cancer" (PMID: 26976969), from March 2016. (The article is freely available online, on the Anticancer Research website.) The abstract of that review article mentions a few different ways that capsaicin might affect cancer, not just as mitochondrial inhibitors. However, they also write that more study is needed before capsaicin can be utilized practically to fight cancer:
While the preponderance of the data strongly indicates significant
anticancer benefits of capsaicin, more information to highlight
molecular mechanisms of its action is required to improve our
knowledge to be able to propose a potential therapeutic strategy for
use of capsaicin against cancer.