You can travel with the old and the new passports. https://www.gov.uk/transfer-visa says:
You don’t have to transfer your visa - you can carry both your old and new passports when travelling to or from the UK instead.
But remember to Update your Biometric Residence Permit!
This isn't covered in your question - but it's important to note that you must update your associated Biometric Residence Permit immediately after a change in name.
Unless you do this you could lose your residency entirely, or be in contravention of the immigration rules. The contravention could lead to you being refused citizenship down the line - as you could fail the "good character" requirement.
https://www.gov.uk/change-circumstances-visa-brp/youre-in-the-uk-and-have-a-brp says (emphasis mine):
You must immediately report any changes to your:
name (eg because you’ve got married)
You must apply for a new BRP within 3 months. You may be fined up to
£1,000 or have your stay shortened if you don’t.
Additional Notes on your old Passport
This is also not covered in the original question, but is possibly relevant. The old passport (in your maiden name) should have been cancelled by the issuing authority when you received your new passport. This normally involves cutting the corners off, and possibly punching holes in the cover.
If your old passport is not officially cancelled, you will likely encounter problems if/when you apply for naturalisation as a British citizen. This is because the UK Government will not allow you to have multiple active passports with varying names in them.
If your old passport is un-cancelled, you should contact the issuing authority and get them to officially cancel it. Make sure that you request it back after cancellation though! They may destroy it unless you specifically request it's return.