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Good afternoon, as she is switching employers it would not be advisable to apply for ILR yet and await until her new tier 2 is issued then apply for ILR.
She would need to await 12 months before apply for British citizenship.
You would need to be under the partner route then with British husband can directly apply when receive settlement.
Once you obtain ILR depending on how long she has been U.K. will determine when she can naturalise as British citizen.
To clarify your daughter would automatically get British citizenship once you get ILR then she can apply for naturalisation.