WGBH does not air BBC programming, however they are working with the BBC, along with WNYC/New York and Minnesota Public Radio to launch a new public radio morning show that's targeting younger listeners.
Close, but WGBH co-produces "The World" with the BBC which airs weekdays 4-5pm on WGBH. They also air BBC World Update 5am-6am weekdays.
WBUR, of course, airs a ton of BBC World Service programming: overnights until 5am and 9-10am weekdays, too. Also 4-5pm on weekends.
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Joe, were you listening on an HD Radio or an analog radio? There should be more than a 2 second delay between WGBH and WBUR on an analog radio at the moment. That's because WBUR delays their analog feed by a little more than 7 seconds to time-match their digital feed. WGBH does not, which makes for (as mentioned) really schizophrenic listening on a HD Radio! I've heard they'll fix this problem after they're done moving into the Brighton Landing facilities.
If you're listening on two different HD Radios, though, there can be noticeable differences from radio to radio (nevermind model to model) in decoding an HD Radio stream. I've had three different models all on the same antenna, all tuned to the same station, and all three were putting out audio about 1/2 sec apart in delay. It's just the differences in internal circuitry.
There also can be delays from digital processing in the different STL equipment that WBUR uses vs. WGBH. WBUR also runs a seven-second delay on their talkshows and, separately, they have a
25/7 ATM (automatic time manager) to "squeeze" more minutes out of every hour during fundraisers and other live broadcasts. The latter is really remarkable technology...you can speed things up by over 12% and it's not audible unless you're REALLY listening hard. That means you can fit an extra 7 minutes into every hour...very handy for fundraisers. I don't know if WGBH has one of those 25/7 devices but I don't think they do in Allston...they might install one in the Brighton studios, though. (shrugs)
To go even further, WBUR and WGBH don't always air the same segments from NPR's national feed of Morning Edition and All Things Considered. Especially for ME, WBUR will shuffle around and drop some segments to make room for locally produced stories. WGBH doesn't do that nearly as much. Sometimes they air different satellite feeds of the same show; this is especially noticeable on Marketplace at 6:30pm when they both air the "same" episode but sometimes the stories are in different orders. That's because APM feeds several "live" feeds of Marketplace starting at 5pm ET, and they mix it up a little for each subsequent feed.