It would also be possible to hitchhike on cars, swimming across the channel is also an option because it’s free.
Asking in case I decide to do this
I would say, in all probability an actual attempt to do that would take the rest of your life.
I actually used to know a guy who tried to walk from the UK to India with the plan to spend no money, although even he did carry an emergency fund and a passport. He spent 4 weeks, barely made it to France [1], ran out of food, no-one would help him, and he didn’t speak French. He had to phone his Mum and ask her to get him a flight back. What you’re suggesting is many, many times sillier.
not actually sure how he made it across the channel, I think by finding someone who had a spare ferry ticket - but dozens of people drown crossing it in a boat every year, people swimming it do so with a support boat, just giving it a go will get you killed ↩︎
I think this is a major issue. It would be really difficult to travel without money in a place where you don’t speak the language. It would greatly increase the difficulty of hitching rides, asking for money or food, and finding shelter at nights. It would also increase the amount of attention you get from the local authorities.
Are you an exceptionally well-trained endurance swimmer, in outstanding physical condition? Do you realize that the English Channel, at its narrowest point, is still over 20 miles wide? And that, without any passport or paperwork, you will be stopped and arrested (because unregistered attempts to swim across the Channel are illegal), assuming you don’t drown.
Even “hypothetically,” and as unrealistic as the rest of this trip is, that bit is particularly silly.
I expect the Chinese government would not allow travel from Hong Kong without a passport or other documents.
You could always try jumping…
The quickest way would be to start by getting a job in Hong Kong, and then saving up the money you earned from it to eventually buy a plane ticket.
I have heard of people doing things like bicycling thousands of miles, but it was never because they were in a particular hurry. If you are, the correct answer is to telephone someone who can wire you money for a replacement passport, plane tickets, etc. That is the technique Jack Kerouac used when being a migrant worker lost some of its initial thrill.
The YouTuber GeoWizard does various nutty expeditions, such as traveling completely across England, Scotland or Norway in a completely straight line, traveling long distances without using roads, or (relevant to this thread) traveling long distances with little or no cash. However his missions are typically a few hundred miles at most. I don’t see how you’re going to cross the vast expanses of Asia on foot with no resources. Every day you need food, water and probably shelter, none of which is readily available at all in some areas, and certainly not without money.
I’m picturing a one-person boat, traveling along the coastline halfway around the world, going down the China coast to Vietnam, around SE Asia, the Indian sub-continent, the Gulf of Hormuz and the Red Sea (a little risky politically), then around Africa, and up the coast and you’re home free. You could live on seafood and seaweed (that might prevent you getting scurvy, I’m not sure). Of course, you’d have to buy the boat and initial equipment, but maybe you could get someone to sponsor a livestream broadcast of your adventures.
Yes, and Chinese Prisons are not nice places.
Now, crossing Europe- that could be done. But not when you have to go into places where “human rights” is an oxymoron.
Not as stipulated by the terms stated in the title.
I dunno about the swimming and hitchhiking stuff, but one condition specified in the OP is that the journey be made without money.
So I propose a way to do it for free: extradiction.
Commit a crime in Hong Kong and get caught by the Scottish police. The easy way might be cyber crime, defraud a bank, or hack a top secret military base in Scotland.
Or maybe murder a Scottish diplomat in Hong Kong.That would make for better ratings on TV. Or a bad sketch on Monty Python. 
Stow away on a boat from Hong Kong to the US, hitchhike across the US, stow away on a boat from the US to Britain, hitchhike to Galloway. Sounds easy enough.
Nice idea, but there’s no way to ensure you get extradited to Scotland specifically and they don’t have separate diplomats. Although there is a Scottish police force, overseas crime like that would probably be prosecuted by the UK government, not the Scottish government; they’d likely wind up in England.
Besides, the OP specifies Galloway; although there is a men’s prison in Galloway, I don’t think they let you choose.
Start digging…
What are you going to do for food?
Perhaps some boats do not scrutinize documentation too closely when hiring ordinary deckhands?
I strongly doubt that a container ship would hire an undocumented crew member in China for a ship bound to the United States without it being an issue with the Chinese police. Probably with the American police as well. It would be such an obvious route for political dissidents to flee a country if they looked the other way when this happened. And by the same token it would be an obvious route for illegal immigrants to enter a country if the destination country looked the other way.
I’ve never been to Hong Kong or elsewhere in China but suspect access to container ports and container ships is controlled and ID is needed. I really can’t imagine surviving long before coming to the attention of the authorities. Also, it’s a famously expensive, crowded city. Where exactly would you be living?