An asylum seeker would make contact with government officials at the border. So, yes, it would mean they would have to enter the country first. It does seem hard to enter the U.K. that way without using a boat/watercraft as the U.K. is an set of island nations.
As far as how 20,000 refugees per year being a "migrant crisis" I do not know. What is a migrant crisis? How many does it take or is it a matter of having enough tax payer funder resources to hand? I don't know. It is usually the second part of the question that stirs up the hornets nest.
I am speaking in broad terms, but a lot of the issues in the Middle East are as a direct result of U.K. meddling. Specifically the promise England made to both Jews and Arabs living in Palestine they would be given sovereign nations if they helped overthrow the Ottoman Empire back in WWI. Not all refugees are from the Middle East. Many are.
What it all boils down to is whether or not major, economically thriving nations are going to be World Police or the World's Motel 6 or both. I think the people of Earth have a lot of war fatigue. So if first world nations won't send their children to die in foreign lands to bring stability I guess those same nations will be importing refugees from those part of the Earth. The real issue is the tax payers in those first world countries are saying "Not my circus. Not my monkey." Another way of saying that is "It's their country. Let them fix it." Yet another way of saying it is, "Let them die."
These countries in conflict have super corrupt governments. They export their poor to the U.S., the U.K., etc. and then take billions in handouts from these same countries. I see only 2 solutions -
1. Send in the Marines. Let the first world nations obliterate these corrupt governments and occupy these countries until the locals get it right and fix it.
2. Take the refugees, but cut off all aid. Who is coming from these countries? Old people? No! It is young able bodied people. These countries are in danger of extinction because their citizens of child bearing years are coming to the U.S., the U.K., etc.
I am not advocate of war. #2 sounds like the better solution. I do realize the financial strain on countries will be huge, but it sure beat sacrificing our kids on the alter of war.