On the BBC logo itself - I still think it is perfectly fine, and has stood the test of time. The channels, however...
I think as much as BBC One, BBC Two, and BBC Four are in desperate need of a fresh new look.
Exactly right - there just needs to be sufficiently strong concepts behind all of the identities. BBC1 lacks any great strength. BBC 2 has stepped backwards and been damaged somewhat. I don't think they can afford to let their channels fall any further.
Even though BBC 1 arguably just pulled off the edit-based refresh to the idents, the gains were short-lived. BBC 4 is marginally the most stale on account of years in use by this point - however its idents sort of sit there quietly and (comparatively) understated... So I actually don't think they grate anywhere as much as some of BBC 1's sequences do, after as many plays as now.
BBC 2's cancelled refresh over a few years ago - an ill-judged 'phase two' for the look - was always going to be doomed because the ideas that were lined up were fairly terrible. It's looking tired now, but frankly was mistreated more than it ever should have been.
BBC 3 is not mentioned - but IMO that's probably the one with the weakest idea and execution visually... I've never liked it. Not as stale, but I just think a couple of those idents look plain messy. It's the use of CG to create entire scenes that just don't look good in CG. Some of BBC 2's 2001 sequences looked better - but then the simplicity of the scenes meant they were both easier on rendering, and looked more clean, more 'ident-like' by a mile. So there's nothing wrong with using computers, they can be vital sometimes - they just don't always work for creating certain complex pieces.
There is also a recent worrying trend of promo designers in the BBC not knowing the correct logos or typefaces, particularly on some network BBC 2 output.
In an ideal world I would have to say, maybe consult someone like Martin Lambie-Nairn, and carefully lay plans for a rebrand of all four main BBC channels, a la ITV in terms of roll-out - as well as re-installing the practice of ensuring logos, icons and typefaces are kept consistent and correct across BBC output into the future. It would take time to do all of this - but to really get it right, I think it would have to. Next best option could perhaps be to rebrand BBC 1 & 2 identities considering some of their stronger previous attributes (not exactly 'back to basics', but sort of) and developing the new look from there.