As nearlythere points out the law is slightly different in England and Scotland. There may be a need to apply for Building Regulations approval as this is likely to be considered to be a change of use of the building.
In terms of fire safety the Law does not make specific rules in the way that you ask. Wht it does say is that you must carry out a risk assessment to identify the general fire precautions required in your building. The guidance documents in the link provided by Anthony tell you how to do this and help by providing examples of different buildings and illustrating what is normally required in various types of building.
The risk assessment covers fire safety from all angles- measures aimed at preventing a fire occurring in the first place, measures to contain the effects of a fire and protecting the escape routes if a fire does occur, measures to detect and raise the alarm of fire, lighting of escape routes, provision of fire fighting equipment to give people a chance to stop a small fire becoming a big fire, signs and notices so people know what to do in case of fire, testing and maintenance of equipment.
What is required as a result of the risk assessment depends entirely on the height, size, character, construction and layout of the building.
So back to your question- it is most likely that the outcome of a fire risk assessment will identify the need for doors to rooms that open onto the staircase to have a fire resistance of 30 minutes, to be provided with fire and smoke seals and be self closing. Many old "Character" doors can be upgraded to an adequate standard using special intumescent paints and varnishes. It depends on their condition and construction.
For an example of such products see
http://www.envirograf.com/acrobat/038.pdf but their are other suppliers of such materials. Be warned however that upgrading a door can be more expensive than to fit a new doorset "off the shelf"
