The Williamsburg Bridge is the other bridge that still carries heavy rail across the East River. It has four lanes of car/bus/truck traffic in each direction, and two pedestrian/bike paths that merge at the Manhattan end. There is currently only one bus that crosses the bridge, the B30, which is essentially a shuttle between Williamsburg Bridge Plaza and the western end of Delancey Street. At Williamsburg Bridge Plaza there is a bus terminal where five other bus lines meet up with the B30. There are also various private buses that cross the bridge. To my knowledge, there are no express buses.
If there were a reliable BRT corridor in Manhattan, the five bus lines could definitely be extended west across the bridge. On the Williamsburg side there are not very many suitable corridors for BRT. The only high-volume one is the short highway connecting the bridge to the BQE, but that could be used to bring buses from Queens to Lower Manhattan. The next largest corridor is Roebling Street, which is four lanes south of the Bridge, but only for three blocks, and then it connects to a number of two-lane streets.
On the Manhattan side of the bridge there are a number of streets - Delancey, Allen, Chrystie and Forsyth - that are way too wide for the amount of traffic they see, and are dangerous at that width. Perfect for the traffic calming effects of xRT. Going north, buses can connect to the First/Second Avenue BRT. Going south is a little more difficult, because the big streets don't actually lead to Lower Manhattan, where the jobs are. They just end at the FDR Drive. This is the same problem that the First/Second Avenue BRT is facing, though, and they're probably planning to take over the roadway under the Drive for a busway, so the buses coming off the Williamsburg Bridge can use that.
However, there is another option: the invaluable James Brennan has a page about the abandoned trolley station under Delancey Street. Some of it can be seen from the platforms of the Essex Street subway station, but he's got pictures. This is a perfect place for bus passengers to transfer to uptown F trains or downtown J, M and Z trains.
Here's the best part: the buses don't all have to stop there. The NYCSubway.org Downtown Track Map (261MB PNG file) shows that there are two unused trackbeds running from that station almost to Chambers Street, that could even be hooked up to buses coming off the Manhattan Bridge. If the tracks were re-aligned so that the subway trains only used the two westernmost platforms at Chambers Street (the M would have to terminate at Broad Street instead of Chambers), then the buses could turn around there. Isn't that fantastic?
There is also the possibility of through-running buses onto the Brooklyn Bridge from that point, just as the trolleys did on the old loop. The northbound buses can be through-run wherever the First/Second Avenue busway goes, but through-running to New Jersey through the Holland Tunnel doesn't seem very promising, because of the number of narrow streets.