A straightforward basement conversion can be completed in a matter of weeks. Converting a single room cellar/basement using a membrane lining system can take just two or three weeks, including digging out the sump and gitting the pump system.
If you need a to convert and extend the cellar beneath the entire house which involves underpinning of the existing structure, this is likely to take several months.
If there is direct access on to your garden or the highway for the removal of soil etc. and your existing ground floor is suspended by timber, you should be able to continue living in your home whilst the work gets underway. If however the ground floor has to be removed and rebuilt because it is concrete and all of the waste has to be carried through the house, you will probably need to move out whilst the work is carried out
If there is adequate headroom then turning an existing cellar / basement into extra living space would cost similar to a loft conversion, approx. £1000-1350 per square meter. It's only when you have to start lowering the floor level to increase the headroom area which involves digging out the ground beneath the house and then underpinning the foundations, that the work becomes more expensive at around £2000-4000 per square meter.