The people who fall through the cracks are the
unhoused, and who sleep rough each night -some people actively avoid housing due to unseen mental health issues, addicts, or being on the run (such as spousal abuse victims, illegal migrants etc). This population is about 5,000 by last count, for a city of 9 million, plus it varies as a significant proportion are only temporary.
However, once again you won't see much of them. Begging is illegal and any who do it (you'll spot some around train termini and outside popular tube stops), are soon hassled on by community support officers, or even charged by the police. Those who keep at it in a constant cat and mouse, are desperate (drug users) or professional fakes, insofar as they are actually housed and earn a pretty penny each day -I recently discovered the fella outside my workplace, who we've known for years, has disappeared due to a community order -he'd been earning hundreds a day and has a house in flipping Chelsea. Some do fake 'announcements' from carriage to carriage on the trains, to avoid the constant copshop, asking for enough to pay for a hostel for the night and a warm drink -they're in it for the money and earn quite the mint, especially given that hostels and meals are actually free. A telltale sign they're addicts or in business.
People with mental health issues are spotted early -treatment and housing takes them in. Also many of the unhoused dress and act as normal as anyone else -when a soup kitchen does pop up, it's quite a wonder that many in line are decently dressed and would otherwise have been invisible. Also excess food from restaurants can freely be given, Chinatown right in the centre's a safe bet -plus clothing from the charity shops on every High street -just walk in and ask. Every supermarket has an aisle or line of trolleys after checkout to donate locally -overflowing each time.
There aren't homeless encampments, though there's one close to it at Marble Arch where the Romanian Roma arrivals land. Once again it's pretty discrete -the tourist site is in the heart of moneyed London and manicured, and the unhoused sleep in a hidden tunnel to one of the high end, underground carparks.
In the past London operated a vast network of beds-in-sheds, in which illegal landlords housed desperate people in outhouses in their gardens, now thankfully demolished after council crackdowns. I've seen occasional tents in Tottenham Court Rd, but only at night and they're gone by day, as it's a busy thoroughfare, a step away from one of the city's busiest restaurant districts.