I got Nottingham based job. I’m living in London. Do I need to relocate Nottingham, or can I work from London office?

I am assuming that London was not an option on the job advert that you applied for, this means that as you applied for a job in Nottingham that the role is based there and you would have to relocate. I would double check the job advert and speak to the vacancy holder. If London was not on the advert then the recruitment would not be a fair and open competition if we then allowed successful candidate to chose other locations that were not included in the job advert. Sorry that this is probably not the response you were hoping for.