Under the Regulatory Reform (Fire Safety) Order 2005: (Known as RRO)
The responsible person (The landlord) has to do a Fire Risk Assessment. This should point out the measures that are necessary to protect 'relevant persons', in this case the persons living in the flats. The RRO does not apply to the actual flats, but the occupiers need protecting from a fire elsewhere.
To give the landlord an example of what could happen: If I inspected somewhere where people were sleeping with no separation between dwellings or the common areas, and with no decent standard of alarm it would be likely that I would end up aiming towards a prosecution (Under the RRO), plus enforcement notices to bring it up to a reasonable standard. There would also be the potential to prohibit the use, but being only 2 storey he might about get away with that one. (Depends on whether window escapes are feasible, as Kurnal pointed out)
The prosecution would probably be linked to a lack of sufficient means of giving warning, with unsuitable means of escape (Due to the lack of fire doors) and the lack of measures to restrict the spread of fire (Lack of compartmentation).
Even with the fire doors, the compartmentation between rooms/floors needs to be sorted out. Lath and plaster does not give enough fire resistance, so everyone in the building needs warning in the event of a fire in anyone elses flat. This is what the risk assessment is for, this would make it clear who is at risk, why they are at risk, and what can be done to get rid of or minimise that risk/protect the person(s).
FWIW, even with a decent alarm, if I turn up with my jack boots on I may still have a potential prosecution purely on poor means of escape and a lack of compartmentation.
The landlord needs to bear in mind that FRS's throughout the country are gearing up for more prosecutions. The days where we would turn up to find a dangerous premisesand just issue letters asking you (Or telling you) to put it right, and generally being helpful, are going. People need to be getting their house in order.