The housing association's policy is a few steps downstream from the primary legislation cited. As owners of the common areas of a high-rise building, they have special duties about safety, which include making sure people are able to evacuate in the event of a fire. This is covered by section 84 of the Building Safety Act 2022 -
(1) An accountable person for an occupied higher-risk building must take all reasonable steps for the following purposes—
(a) preventing a building safety risk materialising as regards the part of the building for which they are responsible;
(b) reducing the severity of any incident resulting from such a risk materialising.
According to 84(3) they must also "act in accordance with prescribed principles", which means that they are meant to follow any official guidance from the building safety regulator.
This overlaps with duties under fire safety legislation which include that a responsible person must "take such general fire precautions as may reasonably be required in the circumstances of the case to ensure that the premises are safe" (8(1)(b)), such as "measures for securing that, at all material times, the means of escape can be safely and effectively used" (4(1)(c)).
Naturally, all buildings are different and what is "reasonable" will also vary. The Health and Safety Executive, which is the building safety regulator, says in its guidance:
What is a reasonable step depends on your building’s individual circumstances and risks. The actions you take should be proportionate to the building safety risks and account for the cost and practicality of carrying out the measure.
It goes on to give an example:
Means of escape are the evacuation routes and exits used by building occupants to reach a place of safety in the event of an incident. It is vital that residents can always use these routes and exits safely. [...] For existing occupied buildings, make sure the means of escape still work effectively. This includes:
...
- ensuring corridors are kept clear
...
This does not mean that there can be absolutely nothing in a corridor, ever. In context, it does mean that the corridors should be kept clear enough to allow evacuation, and that they should be monitored for hazards. This is not a zero-tolerance policy, but a standard "assess the risks and take steps needed to avoid or mitigate them" policy. The physical geography of common areas would determine whether a particular Christmas tree (or anything else) is or is not an obstruction. In front of a fire door? Bad. Tucked into an alcove? Better.
Indeed, it is odd for the housing association to describe their tree policy as "zero-tolerance", given that the tree is there during the day. That is a non-zero amount of time. Their position seems to be that the caretaker's presence makes a difference as to whether the tree would slow down evacuation. They must be imagining either (1) that in the event of a fire, the caretaker is expected to grab the tree and swiftly bundle it away somewhere in order to clear a path, or (2) that the caretaker can help to guide residents safely around the tree, but if he is not there then they might not be able to manage it themselves. I suppose (2) is more likely, especially for a housing association block, if their risk assessment has determined that some residents need additional mobility support.
(As a final note, a Christmas tree can be a fire hazard in itself, but nothing in the article indicates that there is a concern about the flammability of this tree. There are other relevant provisions of the fire safety regulations which apply in that regard. Since the news article doesn't go into that, I won't either.)