Not sure about the DWP part, I'm afraid, but I can help with the BRP and passport bit:
The name on her BRP MUST match the name on her passport at ALL times.
In order to change the name on her BRP she would have to renew her US passport in her married name and then within 3 months of receiving the new US passport, she would have to apply to change her name on her BRP. If she doesn't do it within 3 months, she can be fined £1,000.
I believe she holds ILR, correct?
For holders of ILR, it costs £229 to change her name on her BRP and it can take up to 6 months to be processed. Normally the £800 Super Priority Service would be available for a faster decision, but it hasn't been reinstated yet.
See here for how to change her name on the BRP:
https://www.gov.uk/change-circumstances-visa-brpWhich links to this page:
https://www.gov.uk/biometric-residence-permits/replace-visa-brpThe best time to change your name on your passport is actually just before you apply for a new visa... because as long as it is done within 3 months of when the visa application will be made, there is no need to change the name on the BRP as well... you simply apply for and have the new visa issued in the new name.
Last of all, when it comes to citizenship, can she apply in her married name or will it need to be in whatever name is on her BRP/US passport?
She must apply for UK citizenship and her UK passport in the name on her passport and BRP. It used to be that you could choose which name you applied for citizenship in, but they changed it a few years ago.
So, if her passport and BRP are still in her maiden name when she applies for citizenship, then in order to get everything in her married name she would need to:
- apply for UK citizenship in her maiden name
- apply for UK passport in her maiden name
Then once she has her UK passport in her maiden name, she would then need to:
- change her name in her US passport
- change the name in her UK passport