I've had the same post-Brexit vote conversation with numerous Austrians, all of them travelled, all English speakers, aged from twenty to sixty, and they have repeatedly identified three factors that make the UK uniquely attractive to economic migrants.
All non-Austrian residents must officially notify their presence and location to the government for every day of their stay in the country; this is the reason for the standardised hotel registration form, and even if you are staying with friends, in a rented farmhouse or in your own house, you must submit a notification to the local council. For EU nationals, you can stay for up to three months; after that, you need to obtain permission for residency, which will need proof of income / resources / employment, proof of medical insurance and proof of identity.
None of this applies in the UK. Once past border control, anyone coming to the UK is lost to official knowledge, EU nationals can stay indefinitely with no further proof or registration and the government has absolutely no idea how many visitors / migrants are in the country or where they are.
Here the number plates are issued to the driver, not the car. They fit easily with little spring clips, and you can use the same plates on up to three cars, paying insurance only for the most expensive. Plates are issued by insurance companies through local offices staffed with clever and capable office ladies who scrutinise carefully all documents and make it virtually impossible for a non-qualifying driver to get a pair of plates. Only EU or international driving licences work here, and you still need insurance to get a pair of plates. This means you will see in the Summer in the cities loads of gloomy migrants on bicycles. Mostly they have to be taught to ride them, and it's a comic sight to see a squad of former Sryian tank commanders, Iraqi tanker drivers and Afghani taxi operators wobbling around a sports centre car park on learner bikes. Being confined to cycles and watching Austrian women confidently whizzing about in cars really pisses them off.
In the UK of course you can buy a car compete with plates in the pub, and drive it about without any insurance unless you're caught by ANPR cameras or have to abandon it after an accident. The ability to own and / or drive vehicles with hardly any checks in the UK is believed here to be a major pull.
An engineering manager here had a slight accident when on holiday in the UK and was surprised that he was treated at every stage without having to produce a single document, form or proof of reciprocal insurance. He asked several times with whom he should establish such details, and was told we didn't bother with that sort of thing in England; all were treated, whoever asked. This was a few years ago; it was such a stunning omission that he has queried it since several times with others, and also with me. I confirmed that what he already understood was, in fact, the case; the NHS is filled with deeply humanitarian, caring people who genuinely see no problem in providing treatment completely free of charge to anyone that asks.
Here, your insurer (cost about £100 a month for me) provides you with an e-card that entitles one to treatment. You can use any GP you like but the first thing the receptionist does is scan your card to make sure it's live and access your history. Doctors are paid by the insurance firms per treatment. There are special registration cards for migrants, whose costs are borne by the State. However, treatment is 'care and maintenance' only - any migrants needing expensive or lengthy medical treatments are better off in the UK. Again, lack of checks are felt here to be a major pull.
Of course, many diehard Libertarians may ruffle their feathers and huff that they'd rather have millions of migrants in the UK than have any retreat on personal liberty, but my own experience with the restrictions here is that they're not at all onerous. Austrians also have such a deep commitment to personal privacy that not only is Google's streetview banned here but so are CCTV cameras and drones that can view your land and property, let alone government cyber-snooping as undertaken by GCHQ. I'm starting to believe that just a few simple tweaks to UK systems would reap huge advantages.