ippy,
In a city as large as London what would be the odds be on that happening?
Not outlandish, especially given the proximity of many law firms to each other around the law courts.
The point though is that you're essaying here a variant of the lottery winner's fallacy: the lottery winner thinks, "Wow, I won against 14 million-to-one odds, that's remarkable" whereas Camelot calculates the odds of someone winning as pretty much one, only they don't care who that winner will be so all
they think is, "so what?".
Similarly, had you thought over breakfast, "I wonder what the chances of running into the other secretary are today", the odds would be fairly long. What actually happened though was that you wondered at it only
after the event. You could just as well have bumped into your uncle Fred who emigrated to Canada 30 years ago, found a fifty pound note on the street, or spotted a ring in the grass that your wife had dropped the previous summer. Very uncommon things happen all the time - try dealing a deck of cards and calculating the odds against the sequence that you get for example - but the danger is to assume that there was some special design or purpose beforehand when in fact what we tend to do is to retro-fit significance to the event.
Or to put it the other way, when you saw the other secretary you were the lottery winner and the universe was Camelot.
That incidentally is why the OP is so daft: the odds against the sequence of a randomly dealt deck of cards is 52!, yet any kid can deal a set of cards and produce a sequence of the same improbability perfectly within any normal mathematical expectation.