Now that that rant is over, let’s talk about what you can grow on your side of the pond. I’m not the most fervent proponent of lawns, but people like what people like.
Yes, you can have that kind of lawn. The 2 most common species of grass used in the UK are Perennial Rye and Bentgrass.
Perennial Rye grows taller and more vertical in clumps. Very dark green and shiny with medium-width blades. Best mowed at 3-4” tall. Fairly fast-growing and needs to be mown once a week.
Bentgrass, Common or Creeping, is a finer-bladed grass that is lighter green and spreading, especially Creeping Bentgrass. It can be mown between 1/4” and 2”. If you want to play Croquet or Lawn Bowling, this is the grass for you. To keep it at 1/4” it will need to be mown at least twice a week. At taller heights it can be mown once a week or even every other week.
For any kind of lawn, either organic or synthetic fertilizers are available and with a regular schedule can keep the grass growing nice and thick.
There are decent battery-powered mowers, strimmers, and blowers that can keep the lawn trimmed nicely without exhaust or a lot of noise.
I’m not familiar with pests and diseases over there, but being the gardening-obsessed nation you are, there are lots of nurseries that can help with whatever problems you have.
Oneeye is right that keeping a lawn looking picture-perfect is a lot of work and takes significant inputs of various products, but nothing feels better on bare feet than a nicely trimmed lawn.