What I read was that, even with tubeless tyres, you must run tubes on Morris rims. If you don't, the tire could collapse in a hard corner. My wife won't drive the car gently, so I'm sure hard corners will happen.
Maybe what I've read is wrong. I've worked on 1940s-1960s American cars and haven't had to deal with tubes yet, so I'm new to this.
For example, this is what the Morris Mania site has to say on the subject (emphasis added):
Tyres
As described above, the
standard Minor rims are not designed for modern tubeless tyres. You can fit tubeless tyres to Minor rims but you run the risk of the tyre peeling away from the rim under hard cornering, in which case if you have a tubeless tyre you'll have an instant flattie! This is not a Good Thing to have mid-corner!
The standard Minor rim usually takes a 5.20x14 inch cross ply tyre, but these are not only getting difficult and expensive to buy, but are not conducive to great handling! Swapping these for the equivalent-size radial tyre, a 155/70x14, will give your Minor a much better feel on the road; it will sit in corners much better (hiding the effect of worn shock absorbers!) and even provide a smoother ride. But a 155/70x14 tyre is pretty narrow for a Modified Minor!
Standard Minor rims can accept a maximum tyre size of 175x14, but this is an odd size in the non-metric world of tyres, so 165x14 is more common and keeps the rim within it's limits. Late model Van rims are slightly wider and can safely take a 185x14 tyre.
Remember to always fit an inner tube to radial tyres - even to tubeless tyres.