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Yes. UK visa dependants will have to pay for visa and NHS fee. Visa dependants will need to pay £490 for the visa. As for the healthcare surcharge, you can check how much she'll have to pay here. She will also need to have her biometric information (fingerprints and a photo) taken - there’s no fee for this.
You can choose either of the two passports.
US citizens can enter the UK without a visa, however, they can only stay for a maximum of 6 months. Any more than that they will have to pay for a visa.
To clarify my previous reply - she can use either passport to apply for a UK Visa, however it would be advisable to use her Nigerian Passport considering you'll be applying for a visa in Nigeria.
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