"London Inn" would be most likely.
I was brought up in the area and often played by the river here. My recollection is that there was an ancient-looking whitewashed building near the the bridge, possibly one of the buildings still standing in the vicinity. Most likely, the original name of the bridge was Bedford Bridge (it was always interchangeable even in my own childhood) and Magpie Bridge took its name from the inn, not the other way round. The Duke of Bedford would have been the source of the name. That's just from memory - you may need to verify these as facts.
