No they haven’t taken payment yet. The tracking said it was signed for on 27 December so I imagine it is too soon to hear anything.
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Yeah, I'd give it a week or so after they take the payment.
It's usually:
1) receive application
2) take payment
3) send out application confirmation
4) send out biometrics invite
5) give biometrics
6) visa processing starts when biometrics received
7) visa issued a few weeks later
You're only at 1) at the moment, so there's still a bit of time to wait.