Why did britain never had Cuirassiers?

The British army did have heavy cavalry equipped with cuirasses until the late 17th century. After that there were units of heavy cavalry, just without that particular item of equipment.

I can find no definitive reason why cuirasses were dropped, but factors such as cost, improved movement without them, and decreasing efficacy as muskets improved, have all been mentioned.

Digressing slightly, but on the same theme, am I right in hearing that one or two British cavalry regiments were unable to fully mobilise at the start of WW1 as their honorary Colonels-In-Chiefs were German?

Haven't heard of that. Not sure why that would prevent mobilisation. Any clues as to which regiments?

I will endeavour to find out, I read it about a week ago somewhere. I believe the Kaiser was one C in C, the other Princess Victoria?

You're right about the Kaiser - 1st Royal Dragoons. This honorary position was not unreasonably terminated in August 1914. See

https://www.nam.ac.uk/explore/international-hussar

I haven't found anything about this delaying the regiment's deployment.

Heard the one about Karl Schauenberg, a German national and iron cross holder, who, having been in the German army in WW2 also became afterwards a highly respected and much liked member of Her Majesty's Grenadier Guards Band no less?
I had the pleasure of meeting him once through a friend of mine who served with the band as a tuba and cello player for twenty four years.
The story is his exceptional music talents were spotted by a member of the band moonlighting as a jazz player in one of the London clubs, he was invited along for an audition and a chat with the then Director of Music who himself had been an involuntary guest of The Reich as a POW.
He was accepted and went on to become one of their top performers.