Have a British Empire which remains based around a mixed settler colony/company axis. As India proved, the smaller company administrators can be more willing to marry into local elites (this may be less the case in Africa, but is certainly not impossible given that there's probably been a noticable shift in attitude at home to create this). Then to deal with falling trade revenue the companies decide to branch out into other area, including industry and eventually jobs requiring technical skills.
The colonies will still undoubtedly lag behind and take the cheapest option rather than what's necessarily the best one, but all it really takes is a handful of entrepreneurs to look at the mills of the north of England, and come to the conclusion that they can make a killing by using the larger labour force in India closer to the sources of cotton production, and undercut Liverpool or Manchester with the finished good. It's still a pretty exploitative mindset, but the difference is this one is exploitative in the same way that the attitude towards the poor in Britain is exploitative, and so has a reasonable chance of following a similar, albeit delayed, course of industrial development followed by some degree of social reform.