Oh wait, it appears there was a block on BBC Two.
Yes, usually on Sunday lunchtimes - https://genome.ch.bbc.co.uk/schedules/bbctwo/england/2003-03-02#at-11.00
Nothing as formal as we have now, though, a lot of it was filling space.
Family Guy wasn't shown on BBC Two until 2005 I don't think.
That's right. Family Guy first appeared on the Beeb in 2005 on BBC2. I'm pretty certain they got this for free when they bought the rights to American Dad (as Family Guy had already been shown, and flopped, on Channel Four), and they showed both of them back to back on Saturday nights - https://genome.ch.bbc.co.uk/schedules/bbctwo/england/2005-10-22#at-22.05
Neither did very well on BBC2 and they were swiftly shuffled later and later - at Christmas they were stripped in a late night slot every night - and after about a year they shoved Family Guy over to BBC3. Initially this was also in a late night slot but finally it took off after a while. They persevered with American Dad on BBC2 for a bit longer, its regular slot for a while was Sunday nights after MOTD2 (they also dumped the final episodes of Arrested Development there) but they later punted that over to BBC3 too.