I listened to your source, but I'm afraid it wasn't very convincing.
The evidence they presented was an ambulance driver who had to go around a modal filter via side-streets to get to a patient who later died. We do not know if the patient died because of the LTN. I am aware of a case where Ealing council failed to delivery keys to an ambulance service - perhaps this was that? That error has now been rectified.
Needless to say, this is a tragedy. However, patients also die because emergency services get stuck in gridlock traffic. What needs to be show is that LTNs in general increase response times; but we were not presented with any data to support this. On the other side, there are examples of ambulances moving rapidly through LTNs, since they do not have to share the space with any other traffic. Remember, London's congestion is so bad that we have an air ambulance service. What we really need is data, and the limited amount we have so far is generally positive towards LTNs. One example we can look to is The Randstad, which is a huge urban area containing many connected LTNs, built from the 70s onwards. It doesn't seem to have problems with emergency service response times.
There's also the risk of missing the forest for the trees here. How many premature deaths occur from diseases of inactivity, toxic levels of air pollution and road accidents? Childhood obesity is around 20% in the UK and even higher in inner London boroughs. If you look at old photos of London, a striking difference to now is children playing safely on residential streets. Ambulance response times are not the only metric we need to keep an eye on.
The hosts like to make sweeping generalizations and talk about the "law of unintended consequences", but I wonder if they understand the law of induced demand?