The application will be based on referring to "settling" in the UK. You are a British citizen, but you are not settled in UK.
I am afraid, its not as simple as filling out an online visa, get it approved and travel to UK. Its a lot more complicated.
Child cannot settle until you prove you have the sole custody of child, and with written consent of father. I have not personally guided anyone before in your exact circumstances, but I will try my best to guide you.
Also during application you must satisfy that you will be able to maintain and accommodate the child without recourse to public fund.
In your situation there could actually be a few other options, and walk arounds. Will require research for for some hours. I do not charge any money, so you will have to be patient if you want free quidance, or you have the option of spending 1000s of pounds on legal advice and chasing attorneys.
Now, to begin, please inform about child's father. Have you got the sole custody of child? Some kind of paper to prove that only you are responsible for child.
What is your profession? Do not want to sound rude, but are you employable and can find work pretty quickly? Have you travelled to other countries or familiar with other countries? If so, please name them. How much money have you saved up? or when you leave USA, how much money will you have available?
Please provide as much info as possible. That might give me a chance to inform you of legal walk arounds. Perhaps I would inform you of two different ways, and then you can merge those two ways to get permanent settlement for your child.
regards