Here's some more 19th century uses of the word 'transpontine' in a South London context. Interestingly two of them feature Westminster Bridge. If the word literally means something like 'over the bridge', we should remember that Westminster Bridge was the only the third bridge to be built over the Thames in London, after London Bridge and Putney Bridge. The original Westminster Bridge, built in the 18th century, was in poor condition by the time of these articles and in fact a new bridge was built in 1862.
At the time of the first article in 1846 the Bridge was evidently closed leading Punch magazine to joke about 'the transpontine people' on the southern shore descending into 'barbarism'
'The Exiles of Lambeth
Lambeth has become a sort of Siberia since the stopping up of Westminster Bridge, for there is now literally no communication between the inhabitants of the northern shore and the transpontine people. All means of social intercourse are completely cut off, and Astley’s Amphitheatre might as well be on Salisbury Plain, as far as there is any possibility of getting to it from any part of Westminster. We have heard of vessels wrecked in sight of port, but here is a place of amusement remaining comparatively empty, with crowds walking within a stone’s throw and unable to get to it.
Lambeth is in a state of utter desolation, and the principal street reminds one of a strada in Pompeii. A civil war might break out and all be over before any one on this side of the Thames could know anything about it. The people are becoming quite isolated from the rest of their fellow subjects, and the interests of civilisation are severely suffering. Already Lambeth is a week behind us in the polite arts, and every day that the blockade continues will send them backward four-and-twenty hours towards the barbarism which it has taken centuries to get out of. We should not be at all surprised at hearing through some circuitous channel that provisional government has been established in the New-cut, and that the whole of the Marsh has thrown off its allegiance. During the stoppage of Westminster Bridge the Lambethites are aliens in geography, not in blood, and we can scarcely expect submission when protection is not afforded (Punch, reprinted in Bell’s Life in London and Sporting Chronicle - 6 September 1846)
Chartists on the bridge
'It would be perhaps difficult to define the precise nature of the political opinions of the Jack Cades and Wat Tylers who swarmed about Trafalgar-square, Parliament-street, and Westminster Bridge, as the expression of their opinions was confined to most discordant yells and sarcastic shouts whenever a band of special constables appeared. And, as these motley groups were marshalled, they presented a somewhat quaint appearance, from the odd jumble of aristocrats and mechanics, making a sort of "Constable's Miscellany." But they, as well as the more practised policeman, did their duty manfully, and promptly silenced the lively vociferations of the rebellious rabble. Soon after one o'clock there was a slight collision, but the multitude at once discovered that they were not to be the master, and the Plebs made a retreat more precipitate than Parthian. The softer sex seemed to predominate, and it really was a marvel where all these women could have come from. They halloed and shouted and jumped about, running to and fro with the most surprising activity; and these chaste Chartists were the most resolute foes with whom the police had to deal.
Lord Frank Gordon, with Lord Adolphus Fitzclarence, and some listless Life Guardsmen, were leaning out of a window, but their uniform did not appear to excite what Horace calls the pus of the people half so much as the sight of the more sombre trappings of the police. The decrepid piles of poor old Westminster Bridge could hardly bear at one time the crowds who thronged over its crazy arches. We expected here to have found it rivalling the glories of the Pont d'Arcole; but it more resembled the Pons Asinorum, and was cleared in a most summary manner by the indefatigable heroes of Scotland-yard, who allowed none but respectable and peaceably-disposed persons to pass over; and thus the transpontine passengers soon became as select a field as after a sharp burst with the Quorn or Pytchley' (Morning Post - Tuesday 11 April 1848)
Some other examples...
'a musical mania seems to have taken possession of the transpontine population'
(Sunday 24 August 1851, Reynolds's Newspaper)
'The piece is wholly destitute of literary merit, and the acting is of that school which excites the enthusiasm of our transpontine public'
(Tuesday 4 November 1851, Morning Post)
'The whole action and tone were, in the highest degree, melodramatic, and would have drawn immense applause from a transpontine pit and gallery'
(Friday 21 March 1856 , Elgin Courant and Morayshire Advertiser)
'They came in groups across Westminster-bridge, from the transpontine districts of Southwark and Lambeth” (20 April 1855, Morning Chronicle - describing crowds during French Emperor's visit)