There are parts around the Tower of London that offer spectacular starting points too but I think are low on loo options so no worse or better than HPC.
1) Tower Hilll Terrace on the West side of the Tower - slight downside is accessing the main road. There is a pedestrian crossing that can be used to stop the traffic safely though. But there is ample space, the Tower, the river etc. but worth checking because I can't be sure there is rain cover. Indeed, please note
Legge's Mount (item 29) which I presume is where Simon keeps his Colnago when out of the country and is just accross the moat from said meeting point.
2) In front of the Guoman Tower Hotel, on the river, just east of Tower Bridge at the entrance to St Katherine's dock, views galore. Rain cover available under Tower Bridge and a couple of places around the Hotel. There's a cunning way to cross the Highway safely if going to Southend.
There are plenty of nearby wateringholes willing to take your money if you show up slightly early.
Yes I have a cheeeky vested interest in Borough, but it is characterful and there are 2 areas with plenty of space which definitely don't get locked up. My feeling is it's not too rowdy there really. Probably similar to anywhere on the southbank, like NFT, which is also good. There are public loos in the station if the market's do end up locked.
Like I said before, it might be interesting just to give an eastern location a go for one of the smaller speedy rides. I don't think you really know how things pan out until you try.
For a map of London loos, look no further... :
http://www.toiletmap.co.uk/