Empire is irrelevant here, as it had very little bearing on the domestic political situation in the UK. Virtually all the British Empire became independent over just a few decades, at the same time that mass de-industrialisation happened domestically. Yet during this period elections still happened in the UK, power changed hands smoothly multiple times, there were no revolutions or coups, and even the IRA was at most an irritant. Despite the loss of empire and the rapid change in the workforce and makeup of the economic output of the UK, it still paid its bills, maintained a nuclear deterrent, integrated a large influx of ex-Empire immigrants, pioneered socialised healthcare in the West, and managed to grow its domestic economy at a reasonable rate almost all the way through. You'd be hard pushed to find another country that was as stable, given the specific challenges over that period.
Brexit was a mistake, and recent governments have been terrible, but don't delude yourself into thinking those are existential crises for the UK. The government will get replaced without much fuss, the UK will patch up its relationship with the EU even if it doesn't rejoin, and the wheels will keep turning.
The manner in which a coup is stopped makes all the difference. It only enhances the position of the government if the government is seen to deal with the coup attempt promptly and effectively. If the government is not seen to deal with it effectively, then even if the coup is ultimately unsuccessful, it undermines the government.
In this case, Putin was not seen to deal with the coup effectively. Wagner was able to secure the Southern Military District headquarters - the main operational HQ of the war in Ukraine - in Rostov-on-Don with essentially no opposition, then drive most of the way to Moscow with only minor skirmishes along the way. If we believe the widely-disseminated reports, then seven Russian military aircraft were lost to Wagner actions, but only one Wagner truck was lost in return. Putin was not seen or heard from for a long time, and reports surfaced that he had flown to St. Petersburg, and when he did appear, his message was not particularly decisive. When the coup was abandoned, it was done not through force, or by Putin publicly facing down Wagner like Yeltsin did in 1991. It was abandoned due to some back-room deal, brokered by Lukashenko of all people, the details of which have not been made public.
The Russian government's internal security apparatus appears incompetent because it did not consider Wagner a threat, even though Prigozhin had been telegraphing his intentions for days (and probably planning it for months). The Russian military appears weak and overstretched, because it could not protect its operational HQ by force. Putin appears weak because he disappeared at the crucial moments, and large parts of Russia appeared sympathetic to Prigozhin even if they weren't prepared to directly take part. The image of Putin being the supreme ruler in charge of everything looks pretty suspect at this point, and Russians know it because most of the events happened live on Russian state TV.
Putin has a long history of wriggling out of difficult situations, and he might still pull things out of the bag, but I think this is the beginning of the end for him.