Properly insulated houses keep things cool in summer as well as hot in winter. I was at a friend's last week in New England (USA). Her air conditioning was broken. But because her house is over a hundred years old and properly insulated, it was tolerable inside almost all day despite the fact that we had a high temperature of 38 and a low of 22. (Incidentally, what can make American summers so gruelling is that during a heatwave in the East and South, and increasingly in the Southwest, it does not cool off enough overnight; it's not just the high temperatures. The higher humidity doesn't help, either.)
Well insulated homes and high efficiency windows make both heating and air con cheaper to run, as well. So while air con wouldn't be economical with a leaky, poorly insulated building, it might well be in a properly designed house. This would also improve the success of heat pumps. (As an aside one of the major papers, either the Times or The Washington Post, had an article a few months back on heat pump conversions in Maine, and how one of the reasons residents loved it was that they could have air conditioning! It's becoming necessary there as summers heat up.)
You can design houses to stay cooler in summer without air con, but your windows need to fully open (no big expanses of fixed glass), you can't put giant windows on south facing walls, you need cross-ventilation, you ideally want higher ceilings. You need to be able to ventilate and cool the house when the sun goes down and the temperature drops and then close it up in the morning. If you can't cool it down at night, then you get an oven.
As for installing air-conditioning, the way Americans do it in newer homes is that both the heat and air-con go through the same set of air ducts (either both are run off a heat pump, or in colder climates, there's a separate gas furnace). Alternatively you can use mini-split units, but central is the most practical in a new build. It would certainly be doable in the UK but you would need to design for the ducting.