Here are some things I do not recommend. I would not try to help by supporting a government policy supposedly aimed at improving things for such children, except perhaps if the policy involves removing existing intervention. I believe collective action is likely to make things worse for such individuals.
For the same reason, I would not try to help by supporting private organisations aimed specifically at helping talented children, though the damaging effects may be less bad than in the case of ‘help’ authorised by the state, and at least such organisations do not depend on the compulsory collection of funds from taxpayers.
I recommend individual-to-individual action. If you become aware of a talented child in a poor environment, consider giving money to the parents, or even directly to the child, or encourage another individual to do so. This might take the form of a capital payment or an annuity, made as a one-off irrevocable gift, perhaps anonymously. If there are associated ethical issues, no doubt ways of addressing them can be devised.
If this idea seems strange, I suggest it is because we have become overly used to philanthropy being collectivised, and overly used to the assumption that a problem is remedied if — and only if — there is a government programme ostensibly addressing the issue. There is also a tendency for the problems of individuals to become collectivised conceptually, i.e. to bias thinking in favour of issues that can be defined in terms of social groups. By focusing on groups and statistics, the specific problems of individuals that are not classifiable into neat categories are liable to be dismissed as relatively unimportant.
Of course, gifting money to an individual always risks that the individual will not spend it on what you intended. This is one of the supposed explanations for why the term redistribution in practice usually means providing recipients with free services (often of dubious value) rather than giving them money.
However, I believe the possibility of ‘mis-spending’ needs to be regarded as an unavoidable risk of giving aid. The danger with non-financial support is that it ends up consisting of what the provider wishes to provide, rather than what would actually help the recipient. This can turn out to be not only useless but harmful.
As far as politics goes, I recommend supporting the reduction or removal of capital taxes.
In a society which insists on compulsory education, the best way for a talented child to improve its chances may be private tutors, or other home schooling. State schools, especially non-selective state schools, should be given a wide berth.
For children not from ‘privileged’ backgrounds, avoiding state education depends on the possibility of preserving savings across generations, whether these be the savings of members of their own families, or the savings of third parties who may wish to support them.
PS: Politically well-informed readers may note that my suggestion is at odds with current Labour Party ideology, which seems to disapprove of exceptional individuals rising into a higher class:
Not one person doing better than the people they grew up with, but all of us working together to give everyone the chance to reach their full potential. [...] We won’t stand for a society in which only a lucky few succeed while inequality and poverty hold back millions. [Labour Party Press Release 7 June 2019]Stripping off the veneer, this looks like an inversion: opposition to mobility, on the grounds that it means someone getting something that not everyone will get.