If you already have a bachelor's degree, BSWs typically take two years. Mind you, I live in Canada, but I do know from American friends that they didn't spend any longer than two years on their undergrad degrees, provided they had already done previous BAs.
Did you know that with a BA in psych you can theoretically also do a two-year MSW? That would open up far more doors for you, and especially in the US, qualify you for jobs that are much better paid. I have both a BSW and a MSW, and I'm very happy to have done my undergrad in social work, but I will be the first to tell you that opportunities were extremely limited at the bachelor's level.
I'm also curious to know why you'd want to practice social work in the States. I ask this because the UK has far better opportunities in the profession, does not have the same licensing challenges as in the States, and the human services/social care field appears to be more respected in terms of pay and scope of practice.