Hi and welcome to the forum.
Firstly, until you know what area you are moving to, it is difficult to give an exact figure for council tax as this varies from council to council. Expect to pay something in the region of 1500-2000 UK pounds a year.
This link
Property in Central London will not only give you a list of all the areas in central London, but also properties to rent. A 1 bed apartment will probably cost around 1000 UK pounds per month plus council tax.
Have a look and then please feel free to ask more specific questions.
Regards
Michelle