Finding a good builder when you need one is a tough job at the best of times. So pity London Mayor Sadiq Khan, who needs to find enough to build 65,000 homes a year.

A week after Khan unveils his London Plan — building on back gardens and all — trust the building industry to go and pour cold water on it. But the Home Builders Federation’s extensive survey should serve as a warning for any authority with serious ambitions for scaling up the pace of delivery for new homes.

It conducted interviews with more than 37,000 workers on over 1000 of its members’ sites in what must be one of the biggest exercises of its kind. Coming during a week when yet more humiliation has been heaped on Theresa May’s Brexit negotiations — and the status of European Union workers in this country is not yet settled — the results are an eye-opener.

On London building sites, 56.3% of all our workers are from overseas, with 49.5% of the capital’s construction workforce hailing from the EU. Among those EU nationals virtually half come from Romania, with Bulgarian and Polish workers in second and third place. And the breakdown of individual trades also shows the alarming reliance on those workers by our housebuilders, particularly in the capital.

In London, 70% of carpenters are from the EU, along with 61% of all labourers and 44% of all bricklayers. Finding a good painter and decorator might also be a struggle in future as more than half — around 54% — come from our soon to be ex-partners in Europe.

By dint of its size and economic dominance as a magnet for workers, London depends on those EU builders much more than the rest of the country; for example just 15% of carpenters across the country overall come from the Continent. And in regions such as Yorkshire & the Humber, less than 2% of building workers are from overseas.

The other huge problem is demographics. Even before the Brexit vote, there was a twofold crisis in attracting new recruits as well as the gradual ageing of the present workforce. The accession of the eastern European countries in 2004 helped a short-termist industry paper over the cracks.

Construction industry expert Mark Farmer’s report for the Government — published a year ago under the stark title of Modernise or Die — predicted a demographic “timebomb” which could see a 20% to 25% decline in the available labour force within a decade.

The HBF’s industry census underlines that warning. The overseas workforce tends to be younger than UK builders as well: overseas workers are much more likely to be in their twenties or thirties than their UK counterparts; 90% of workers above the age of 50 are UK nationals. Many builders are hiring — retraining army veterans in housebuilding skills for example — but that’s barely scratching the surface regarding the labour required.

It isn’t as if the building industry doesn’t have enough issues. Farmer’s report last year highlighted a “dysfunctional” training model — the Construction Industry Training Board has just unveiled a major restructuring in response — as well as a lack of innovation, and “non-existent” research and development, which in turn fosters a low-productivity culture.

Chancellor Philip Hammond’s budget talked about encouraging modern methods of construction — off-site factory manufacturing, for example — through government procurement. But it’s a fledging sector which needs its own skilled workers and in all likelihood a more proactive stance from the public sector to fly.

There are a few silver linings in the federation’s census; for example, less than 10% of overseas workers surveyed say they plan to leave the UK industry (assuming, that is, that they’re allowed to stay). But that doesn’t tackle the potential threat of restrictions on future arrivals; hence the industry urgently needs to find another pool of labour. That could be through the outreach programme to schools recommended by Modernise or Die, although the Government ducked the report’s call for a levy on the developers who rely on the industry to help pay for skills.

Losing the ability to deliver on infrastructure is up there with the biggest Brexit threats, despite all the acres given to City risks; but most alarmingly of all, there are no housebuilding roles on the official “shortage occupation list”.

At the very least, that should keep Khan awake at night.