I live in lewisham can I move out at 16 with out my mums permission

Where exactly, is lewisham (state, country?)?

London South East

From my research I believe UK law allows you to move out at 16.

Are you sure yeah?

I couldn't find a specific law that says so. But everything I found does indicate that. If you want to be sure consult a local solicitor. You can also try your school counselor.

But how will you support yourself if you move out?

Yes you CAN move out at 16 with parental permission.
Although I would question the wisdom of this..

Only WITH permission?