The actual report says that there are 125,000 homes in London without permanent residents. Those homes aren't unoccupied; they're often used for AirBnB or temporary rentals. That's not a vacancy. In reality, only a fifth of those homes are empty most of the time.

Also, the report doesn't tell us how many non-vacant homes there are in their survey area, so we have to guess. London reportedly has 3.59 million dwellings, so 25,000 vacant homes would only represent 0.7% of homes being vacant. Even if you use the higher 125,000 number, that's still only 3.5% of all homes. That's a fairly low vacancy rate, and about right for a city.

Even after the pandemic, housing officials report a 3.4% vacancy rate, which is the highest in eight years. https://lfpress.com/news/local-news/london-apartment-vacancy.... Before that, it was only at 1.8%. That's extremely low.

You can tax vacant homes, but in general, those taxes have a very small effect on housing prices. It's far, far more effective to allow people to build more housing, including luxury housing.