In fact, it's called combined heat and power (CHP) with district heating.

This is something widely used in Europe (especially Denmark and Germany). 30% of Berlin's (ca. 3.5 million people) space heating requirements are covered by CHP.

So, it's absolutely no rocket science and has been refined and optimized for decades already. Just not in the UK or the US.

CHP causes slightly lower electric conversion efficiency (like, 5% less), but makes use of this 5% loss to deliver another like 20%-30% of the otherwise wasted primary energy to the homes.

CHP with nuclear plants is not exactly a good idea. Consider a radioactive leak in the plant, contaminating the hot water loop. It would have to shut down immediately to avoid pumping radioactivity into the homes, leaving entire cities cold from one minute to the next.

Micro-CHP units in individual homes are half-nonsense, because
- the capital cost for a home owner to buy such a thing is much higher than for large-scale utility companies,
- maintenance needs to be organised in much more decentralized and therefore more inefficient way,
- a change of fuel is almost impossible and
- propagation of such units is much slower than with a central solution which would serve whole cities or at least city quarters in one go.

Cheers,

Davidyson