This is a bit misleading. Is the UK 'full?' Well by what measure? We could cram a load more people in, we could build over all the nice green bits in posh places like Chipping Norton, the Midlands, destroy a few national heritage sites and put up blocks of flats. Deal castle could probably be a nice little refugee centre or converted into modern housing. We are objectively not 'full' by standard definition until not another person can set foot on the land and be housed, until we have nowhere else to build homes for them but this is a conflation of what people actually mean.
When they say 'full' they mean we can't support any more people. Our NHS is collapsing under the pressure of (a) austerity; and (b) the pure amount of people. Doctors are striking constantly, GP's are trying to take appointments by phone now, average waiting times are astronomical (anecdotally I once waited four hours with a broken ankle because they forgot about me). They can't afford to give people the medicine most likely to save them (as someone saved by the NHS this is highly egregious to me) and can't handle the current rate of patients. Departments have been placed on alert this christmas because they don't think they'll be able to handle the spike in admittance due to festivities. The police are increasingly failing to prevent and investigate minor crimes, there aren't enough of them to physically commit the manpower to enforce the same levels of safety as a few years ago (although crime is statistically falling so perhaps a social shift has prevented this effect being too bad)
There are 4.8% of the country unemployed at the current time (
https://www.ons.gov.uk/employmentandlabourmarket/peoplenotinwork/unemployment) which whilst a reasonably low figure is a lot of people. Most can't find jobs. The market is competitive and often people have to work two or three jobs to support their homes and feed their children with petrol prices rising, food prices rising (per gram) and the minimum wage simply not adequate to compete. The poorest can't take advantage of globalisation, especially in the North where there's little industry or 'city' work (a la London financers), they can't afford to move to Spain to look for a job whereas if Spanish tax laws are better the company can freely establish there moving more jobs away from the poorest in society. So much production is outsourced for a reason. Workplaces are revolving doors, if you're not good enough you don't get a chance to improve they'll just throw you out because there are ten other people queuing up behind you to take your place.
The situation isn't all that great and keep adding people in to the mix isn't helping. Another body is exactly that. Our population growth is internally rising, keep adding in people from outside is causing problems. It's not about what race they are or where they come from. Common sense should show keep adding people to a cracking system isn't helping. Ideally we need a reshuffle. To slash high end corporate pay packets and force the wealth down the chain but I don't see the tories making good on any promises such as this so we have to look at what can practicably be done.
This is excluding the social factors that are effecting how people view fullness.