Land is only finite in areas where lots of people want to be - like for example your example of the south-east. Finite or scarce resources are normally allocated by way of price, and I see no reason why any further rationing measure should apply.
Or, to put it a different way, I wouldn’t be giving up my 200ft-long garden in a south-east commuter town without a fight, no matter how finite land availability in the area!
To be pedantic, land is finite everywhere. It's just that, in rural areas and places where there is no chance of getting planning permission for houses, demand for the land is not high enough to push the price up very much. And actually, just about everything you might choose to buy has a finite supply. The peculiarity of land is that the limit of supply is determined largely by nature, completely independent of demand, and there's little that any business can do (certainly, in urban areas that are already full of houses etc.) to increase supply. This contrasts with most goods, where the limit on supply is determined by how much businesses are prepared to supply at the price most people are willing to pay: The number of mobile phones in the World is finite. But if people were willing to pay more for mobile phones, then you can be certain that companies would find ways to increase the supply. By contrast, the amount of land in London is fixed, and that amount of land will remain (roughly) the same no matter how much people are willing to pay for it. (Yes I know there will be small effective increases because of, for example, building tower blocks, and securing planning permission to build on the tiny amount of unused land, but that's marginal: The amount of land is fixed to a reasonable approximation).
Throw into that mix that land is something that everyone needs at least some of it: Because everyone needs somewhere to live. And you have a problem that rationing by price generally leads to awful consequences - like people being homeless. Usually I'm a very strong supporter of free markets - because in general they are by far the most effective way anyone has devised to allocate resources so that we as a society derive maximum utility from them. But when you have a resource that is (a) fixed in supply, and (b) essential to life, then I think we should question whether price alone is the best way to allocate that resource.
In the case of land... it gets even worse because you have people owning the land and then renting it out to others who want to own the land but can't afford to. To some extent that means those people are making a profit, not by providing something useful, but by hogging a resource that other people need. Is that an ethical kind of business? Is it something that we should really be encouraging as a society? (And I say that as a landlord myself - although in my case, I let to students who probably don't want to buy their student residences, so I think that's ethical). Of course, on the other hand, like you, I have a garden which I'm quite attached to and would not wish to give up.
Lots of questions there and I don't really know the answers. But I certainly think there is a case for saying that, for something like land, who gets to use each plot of land ought to be determined by something related to social need, not just prices and market forces.