I'm a little late to the party. I hope you're all still around.
Some months ago
The Inbetweeners had its tenth anniversary.
This article describes how teen culture underwent a substantial change very shortly after that program wrapped.
Yes, Minister was very deliberately designed to be timeless, and it has largely succeeded in that.
The Thick of It, by contrast, is listed by TV Tropes as an Unintentional Period Piece. Two years ago Armando Iannucci
declared that it would be impossible to make now because the state of politics has moved beyond parody.
I concur that certain elements of Davies-era
Doctor Who might not fit now - particularly the flatulent Slitheen. I understand, of course, that these things were often a necessity to get the series commissioned in the first place. As early as 2007 it is clear that the program had re-established itself enough to move back in a more classical direction.
As for the spin-offs, I don't think
Torchwood will age well, based on a few episodes that I have watched back in recent years. The presentation and style felt very dated, not to mention the look-at-us-we've-got-nudity-and-swearing problem that some members have brought up against DIS. Perhaps
The Sarah Jane Adventures will fare better in that respect.
Class, of course, was widely regarded as dead before it even began.
That televisual giant
The Simpsons has been accused of this many times in latter years. A lot of the family dynamics (and indeed character designs) are perpetually trapped in the late 1980s.
Sometimes being up to the moment doesn't automatically spoil a production:
Have I Got News For You and
Mock The Week are still endlessly repeated on Dave, even though the particular weeks which they mocked might be a decade and several governments ago.
What particularly interests me is the fate of Clarkson-era
Top Gear. A lot of older episodes won't actually work as consumer advice (because the cars being reviewed have long since gone out of production), but the comedy side doesn't seem to have been affected.