Your assumption is correct. This will not be an easy or a smooth path. And it will happen in a few months. They've been talking about needing 10+ years to deliver their goals. But they need to show enough progress in about 4 or 5.

This was an interesting election.

• Labour won in a landslide, but voter turnout was low (lots if thoroughly disillusioned people).
• A lot of the new seats they won were by narrow margins, indicating that while they have a mandate, it's not as emphatic as the MO count might suggest.
• Regardless of their policies/objectives, Reform made good gains for a new party. Not many MOs to sit for it, but they will feel invigorated by how many races in which they outpaced the Tories.
• The Liberal Democrats has a fantastic night, increasing their MP count by something like x7.
• The Scottish independence cause took a hit with the collapse of the SNP.
• This election was a spectacular culmination of missteps (both individual and organizational) for the Tories. And this might have been a mortal wound for a nearly 200 party that will massively change British politics. Such a collapse hasn't been witnessed for centuries. If they survive it might be as a nationalistic husk within which the Reform party takes up residence. Though nothing is certain for the Tories at this point. They might come back from the brink of death; they might shuffle back to the centre; they might become a third party for the next couple of decades; etc. Labour had to do a huge rebuild over the last 5 years and managed it. But their situation is very different from the Tories, so that analogy isn't the best.

The House of Lords serves as a check and balance against a government running amok. Now, they're not necessarily a good check or balance, but every government needs one. Very occasionally they have been - to be mildly disingenuous - useful idiots. (And occasionally, obstinate asses, but I digress.)

Ideally though, we could do with a House of ... whatever's below Common, because if the ones in the Commons are commoners, what does that make the rest of us?

And how would we stop corruption in this lower, lower house?

But nonetheless, it would be useful for a government to have to take heed of people who are closer to the real world. (And I don't just mean MPs' surgeries or correspondence because the repercussions for falling behind on that are slim at best.)

There's literally privatization proponents in this new labour government.