Just briefly, the legislation for fire is not that complicated. The Fire Safety Order (sometimes referred to as the RRO) covers only the communal areas in your block and so you must have a fire risk assessment for these areas. Because there aren't sufficient employees (5) there is no requirement for the risk assessment to be recorded in writing, though it is always a good idea to record your findings anyway.
If your block was purpose built in 1982 the it is very probable that it is built to such a standard that fire will not spread from one flat to another. If it is built to this 'flats' standard then the block is not a 'house in multiple occupation' (HMO) and Environmental Health (through the Housing Act) need not be involved.
If the block is a converted house and the conversion was not to the building regs standard for flats then it will be considered an HMO and you should contact Environmental Health at your local council.
Anyway, it is probably safe to assume that in your case this is a purpose built block. So what do you have to do?
Certainly try to conduct the risk assessment yourself, you can get guidance here
http://www.communities.gov.uk/fire/firesafety/firesafetylaw/There are downloadable guidance documents and there is a downloadable fire risk assessment form which is very brief and hardly adequate for most places, but for you I think it will be sufficient.
It isn't that complicated - just use your common sense and think about each of the areas I mention below in turn to make sure that you've done every reasonable thing that you can to minimise the risk from a fire.
Your findings are likely to be along these lines (remember this is only for the common areas):
Means of escape - satisfactory provided all flats still have fire doors as their front doors (e.g. none have been changed for plastic doors). Any ventilation that was provided in the past, such as openable windows, should still be in place. Doors should be openable from the inside without the need to go hunting for a key.
Fire alarm - satisfactory. No requirement for one in the common areas. Note, that each flat should have a self contained smoke detector inside it to alert occupants of that flat only (i.e. it doesn't need linking to the other flats in the block)
Emergency lighting - it would be very sensible to ensure that, if the power fails, there is sufficient light on any stairs or steps and at doors to ensure that people can safely find their way out the building. If you have no emergency lighting, is there sufficient borrowed lighting coming from a lampost outside the window? If not then a couple of carefully placed emergency lighting units should cover the risk. Even there is a bright lampost outside the window you should consider fitting them anyway, to cover a widespread power cut.
Fire extinguishers - no requirement for them in the common areas unless you have any specific fire risks within those areas you want to protect.
Signs - no requirement.
Maintenance and testing - look after what you have in place.
Changes - if there are changes to the property review the risk assessment.
While you're doing all this, don't forget to keep on top of any health and safety shorfalls as well - look for damaged carpets and at the adequacy of guard rails and bannisters, etc. Again, common sense stuff.
Have fun
Stu