We have a contract with them that covers services and repairs.
However. Regardless of this we were originally told it would be several days. When they learnt a family member had specific medical needs, they then said that meant we were vulnerable and as a direct consequence of that British Gas said off their own bat* that we were a priority and came out within a couple of hours.
That is exactly what I said, you are very lucky that they agreed that specific repair fell within your contract. We also have a contract, we also had a "vulnerable" resident ( British Gas's definition) however, they decided our repair didn't fall within the contract, due to a law changing which affected positioning of boilers.
In spite of our contract, and in spite of our having never owed then a penny in quarter of a century as their customer, our credit rating was too poor for a payment plan.
so 18 months without heating and hot water, whilst we saved up.
So British gas will move "vulnerable" households up the list of waiting households, (to the detriment of other households who have paid equally into their insurance - obviously)
This is a purely commercial decision.
They do it because they make money out of it, no other reason.
They are not obliged to, there is no law to say they have to, it is in their contract as a selling point only.
Really don't see what bearing this has on the OPs land lord.