The House of Lords would provide a backstop check against any abuse of the new structure by English factions.
This would have the benefit of:
1. Simplifying the process of enacting UK-wide legislation.
2. Airing non-English electorates' opinions directly within the real centre of decision-making (the cabinet) and in a way that bypasses UK-wide party control.
3. Snuffing out the tiresome calls for English votes on English laws and other ways of supposedly rebalancing the UK constitutional settlement. The people clamouring for these gruesome fixes should be pushing for the dissolution of the UK, not for tinkering around with House of Commons procedure. In the short-term, "4-parliaments-1-cabinet" or 4P1C would give England control over its own government without needing any increase in the number of politicians. Meanwhile the non-English countries would be able to use their cabinet representation in order to opt out of any legislation they might not like.
Long term, the trend is undeniably towards the establishment of three independent sovereign countries in GB and a single united republic for the other island to the west. We should not invest much effort, time or expense in the meantime towards shoring up the dying UK.
It's also time to apply to international sporting associations for our remaining non-devolved sports teams to be split up. We compete as separate nations in the Commonwealth Games, Rugby, Rugby League, Football etc; it should be likewise for the Olympics.