There's no rule that says that, in fact the rules in the UK are that you can pay people below minimum wage if they're under a certain age.
I think there are also some tax incentives that are designed to make employers more friendly to the idea of hiring a young workforce - all to offset the "cost" of training people, but of course this is just used by mega-corps to have a perpetual stream of extremely cheap labour for frontline staff. (Argos, McDonalds).
So it's possible that not only do they not pay tax, that also their employment is partially subsidised by tax.
Even if below the income tax threshold, they might still be above the Class 1 NICs threshold (which is £250 or thereabouts).
But no matter how low their earnings, all employees must be registered for PAYE from age 16 onwards and will receive an annual P60, as well as a P45 when they leave.
That is irrelevant. If a 5-year old wins some competition, they will pay taxes on that money. That doesn't mean we should give 5-year olds the vote, does it? Also, plenty of non-tax related laws apply to children - why should they not get to vote on Education or Healthcare, for example?
Also, while some 16-year olds work and pay taxes, the majority do not. So a blanket right to vote is not justified even by this strange criterion.
Under 16s also pay income taxes in they have income. They also pay various other taxes such as VAT. Non-citizens pay taxes but do not get to vote.
The general expectation is that they should be in full time education or training until they are 18.
All that makes "a minority of people in this age group pay taxes" a very weak argument for giving all of them the vote.