Not as much as in the US.
I'm aware they used to provide for US forces in the 2 world wars, but its not something British people really know much about. It is also a mid-range brand. It isn't Rolex or Breitling. People aren't likely to have owned one, (like casio or swatch), nor heard of them because they are an exclusive brand like a rolex.
I actually hadn't heard of them. I was looking at Victorinox watches (they make swiss army knives of course as well as Swiss watches) but I'd seen a lot of reports about them having ropey timings where the second hand doesn't hit the digit perfectly and that would piss me off. The hamilton actually has 4 ticks per second so it looks very smooth. I think Rolex have 9 ticks per second as a comparison. Most cheap watches are one tick.
Anyway, I was looking at other brands and yanks were raving about Hamiltons and I figured they looked cool so I got one.