There are several issues OP which is why it can seem confusing I think.
One is that the flow of refugees has really increased in recent years, not just in the UK but everywhere. The rules for refugees came out of the post WWII period and reflect what things were like then.
Now, there are much larger numbers of people migrating, and they aren't necessarily the same kind of people - they may be more differernt culturally, or in terms of education and need for services to settle. These larger numbers and the differernce in their needs creates a lot more stress with integration, and in some places (like places in the US near the border) there is a huge demand on services that can't always be met.
In addition, a large number of the additional migrants are really more economic migrants, who actually aren't really eligible for refugee status. Having them come and try and claim asylum still costs resources however. And then there are also increasing numbers trying to fraudulently claim asylum, and it's becoming increasingly sophisticated.
The other issue is people coming legally, and the effect this has on communities. It has benefits of course, socially and economically. But it can stress community cohesion is levels are too high. And economically, adding a lot of workers tends to suppress wages and dilutes things like union action around benefits and working conditions. So this can appear to boost the economy, first because these people initially are new consumers, but mainly because it's very good for people at the top of the economic ladder who are employers - they can have cheaper workers used to less advantageous working conditions. It can also be pretty good for middle class people who get cheaper goods and services. But it is not good for the people who are at the same level as the imported workers who find their ability to demand higher wages and better benefits is reduced.
The other issue is with lower waged immigrant workers, that they will not likly earn enough to pay for the services they use, so it begins to bleed money out of public services - this is true even when it's workers that are sorely needed and is a real conundrum.
With all immigration, if the numbers are high, it will further stress any problems with infrastructure and housing and such. If you are in a housing or doctor deficit, add half a million people, you can build that many new houses and gain that many new doctors, and not get ahead.
There are other issues which are important but maybe not as central: the question of whether propping up an economy that way is sustainable; the tendency for governments to avoid investing in training their own people but just get specialised workers from abroad; the ethics of taking the best educated people from poorer societies after they have paid for their training; and the real problems that can arise in communities where you have ghettos form or where there are really significant differernce in cultural values.
None of these are specific to the UK, they are problems in most western countries to a greater or lesser extent, except where they have really restricted immigration.