Question about getting passport in new married last name!

I arrived in the UK on a spouse visa in November - through all the hustle of getting married & moving aboard, I forgot to ensure to legally change my last name in the US with my social security card. On our marriage certificate, I did check to take my husbands last name. Now, my question is since I’m here in UK, how do I go about getting a passport in my married last name? (I’m a US citizen)

Thanks guys! This sub has truly been such a support system and I appreciate any advice! 😊