[Robert Williams]

The underlined BBC logo first appeared in 1987, but didn't start regularly appearing on copyright straps until 1990 (though there's a handful of earlier examples, especially from Pebble Mill programmes), but the 1972 rounded logo still appeared on some programmes as late as 1993 (and I've seen it on publicity photos from even later).

Actually and officially (according to most sources), the underlined BBC logo was launched in May 1988, and making the appearance on some programmes in the same year and news programmes in 1989.

I think it's been stated on TV Forum or TV Live Forum that the date of May 1988 is sometimes given because that's when it appeared on an ITN bulletin.  Clearly, the BBC would not themselves decide to launch a logo on a news bulletin on a rival channel.  In fact before 1997, the BBC didn't really 'launch' logos as such, they just gradually crept into usage, which is why it's difficult to ascertain exactly when the late 80s logo came into being, and why the changeover took such a long time.

As mentioned, it appeared in a See for Yourself supplement with the Radio Times (to be precise, it was the 2-8 January 1988 edition which would have been on sale in the final week of 1987), which is currently the earliest evidence of its use I have so far come across.

The 2021 logo is gradually creeping into use as well, but nowadays as you see BBC logos a great deal on screen and online the slowness of its rollout is rather more obvious, as opposed to the late 80s when you didn't actually see the BBC logo that much on screen, apart from on copyright notices.