As I previously stated Salmonella is a threat to those in a high risk medical and age bracket.
I shall tell you my experience as I think it will help you more from someone who has had a cat test positive for Salmonella.
I bought in a kitten from another breeder, the kitten had runny poop when he arrived - I put this down to stress, but as normal I took him for his vet check after a couple of days. He did a poop just before going to the vets so I grabbed the sample and took it with me for a full panel (as I have a multicat household) The panel covered all the nasties plus I requested TF. His breeder TF tested the litter but I got it done anyway because if anything was going to shed it would be the stress of a new home.
Everything came back negative apart from Salmonella. As you are in the UK by law your vet should have sent the sample off to a government lab as I previously mentioned to isolate the strain. Can you please check with your vet that this has been done and what is the outcome.
The labs will give an advisory to your vet on how to deal with the situation - unfortunately more geared to cattle for example having a 'boot dip' before entering the barn
In reality lets look at how it is spread : Simple - POO! The reason I suggested confining your cats is because they have runny poop, they are going in the litter tray, doing a poop, the poop is not solid so chances are making a mess within the litter tray. They then get bits of litter on their feet and walk this around your house. If the salmonella is present the reality of this is that it isn't great - it can be transmitted to humans and those in the at risk groups are more susceptible - end of.
It is up to you how you deal with this in your environment. I had my kitten in a room, I cleaned and bleached his litter tray daily, I followed a correct hand wash routine exiting the room and wore gloves when dealing with the litter tray & steam cleaned the floor and used a proper disinfectant on all surfaces that kills salmonella, MRSA, Ecoli etc
My 8 year old nephew had been in contact with the kitten and I had to inform his dad to keep an eye on him as he was in the 'at risk' group. I also didn't have visits from him until I got a negative re test.
After 3 weeks I re tested and the Salmonella was gone, he was then retested a couple of months later and still negative. I do routine tests to make sure. I do feel I might have caught the salmonella in a transient state rather then it actually latching on to his gut but I was also aware of the possible implications should it be a serious case.
It is upto you how you deal with it and the housekeeping in your own home.