One member of the Royal Family was left on the Surface. There’s no mention of a Prime Minister in Fallen London that I’ve yet seen, suggesting that Lord Palmerston also escaped the Fall. Unless Parliament was in session at the time, it’s likely that at least some governmental representatives also escaped the Fall.
If everyone kept their heads, a functioning government ought to be re-established within a few days to a few weeks.
Some potential changes:
The Empress’ Shadow might find herself ruling in fact instead of mostly in name.
Scotland, Ireland and/or Wales might decide this is a great time to secede.
The British East India Company might find itself with a functioning army, fleet, and bureaucracy at a time when all three were in demand. How would the people in charge use that power? If the people involved with reducing miltary presence in India were stolen along with London, the succeeding government might have taken radically different policy towards the colonial holdings.
Across the Pond, The American Civil War was in full swing and the Confederacy was pinning a significant hope on the gamble the Europe in general and Britain in particular would be coerced into intervening militarily for fear of losing their cotton supply. Historically, that hope was unfounded; Britain remained neutral, even in the face of an inflammatory incident in which a Union ship boarded a British vessel in order to capture some Confederate diplomats that were on board. The “King Cotton” gambit was viewed as extortion, essentially, and they refused to be drawn in by it.
In the FL-verse, the loss of London could have had consequences in both countries. In the Confederacy, it would have changed the strategies and plans of the Confederate government. In Britain, an all-new government with a significant number of new and untested politicians, and influence out of proportion to their ideas compared to the normal circumstance, might actually vote for intervention. Especially if, in the face of conflict in the Colonies, a desperate government suddenly saw benefit to recognizing the Confederacy and developing it as a trade partner.
It just goes to show that you can imagine all kinds of things.