It was tried with 'Ten o'Clock Live', which had a similar format to 'The Daily Show', and had it's moments, but suffered from:
1) Jimmy Carr
2) Lauren Laverne, not really a comedian and, as such, rather out of her depth.
3) A hyperactive schedule and lack of focus, which meant that the kind of stuff like wot you have posted couldn't be done (major politicians brought in for debates and interviews ended up getting shunted off after five minutes so that Jimmy Carr could dress up as a woman and spout some hypocrisy about tax breaks for the rich, etc.)
4) Being scheduled against Question Time, the only politics programme anyone watches.
David Mitchell and Charlie Brooker's rants were quite funny, but the sketches were generally a bit thin (as, to be fair, they often are on TDS). It probably needed some of the flexibility that meant that they could really start rounding on a topic and not cut to ANYTHING INVOLVING JIMMY CARR (did I say that I HATE JIMMY CARR AND WOULD WISH HIM DEAD if this weren't the internet and that sort of thing could be taken literally a la Robin Hood Airport and the blowing up thereof, so instead I'll just WISH THAT HE STUBBS HIS TOE OR SOMETHING AND IT REALLY HURTS). Also, the ability to do bias on US TV helps this kind of thing along. The flip side of stopping a kind of UK Fox News is that no one really says anything interesting; Rory Bremner has said the reason he can't get on TV these days is that no-one wants to upset the politicians( though probably him being not very good anymore and not being able to do Cameron and Miliband any better than a twelve year old boy doing Frank Spencer probably comes into it). In the meantime, HIGNFY is at least funny and probably draws attention to a lot of issues that people would otherwise miss, though it doesn't really solve the problem of the lack of in depth satire. God, I've gone on a bit, people'll be thinking that it's spam. And maybe it is, it's about as interesting.