Apart from death, one of the worst things that could happen could be losing your job or your partner due to your horrendous breath. Bad breath is a HUGE turn-off and working with someone with bad breath is really unpleasant; it’d be worse if your job was customer-facing.
If your partner or HR manager had a quiet word with you about your breath and you said you were doing all you could, they might be sympathetic, but if you refused to do anything about it then you could have serious problems.
I wouldn’t say people “managed” without brushing their teeth for millennia. They lived long enough to make babies, start civilisation etc, but they would have suffered a lot from dental pain and would have been a lot more prone to illnesses caused by dental problems, like the aforementioned cardiovascular disease.
At the British Museum they currently have an extra exhibition on mummies (they have the big permanent one too, this is a more detailed look at eight specific ones). According to that exhibition, when the mummies are scanned there is a strong correlation between dental problems and cardiovascular disease; none of the mummies were of old people. These would have been rich people, so it’s possible their diet actually made them worse off dentally than poor people, but the connection was quite remarkable.
BTW, I thought the false stereotype about the British was having wonky or yellow teeth, not bad breath or lot of plaque. So the joke fails in that respect too. Though I’d better be careful not to sound too annoyed at the joke, or I might get a warning.