Thanks for such a detailed (and intelligent) response to a question that I think most people wouldn't take seriously (although I meant it seriously).
* Airports: How much is it really going to help to build one more runway? What about when another runway is needed, and another after that? The best thing to do is to plan for the airport to take up an enormous amount of space beforehand, and probably, in this case, to build another airport. (NYC already has three, but I only know of two in London off the top of my head.)
* Roads and railways: I think this is the "most difficult" one to deal with. I think it helps to have a more accurate definition of property rights than what most people operate with, which is to say, it's only your property if you actually use it--otherwise nobody is impacting you if they use it, which is the whole point of property. For example, I don't think people should be allowed to "own" large, empty tracts of land. (This would be the common problem with building roads and railways in America.) And if you want to build new roads in a heavily urbanized area, I think that's just a bad idea... I mean, that's never really necessary, unless you want to increase the human density of that particular area (which it self is never really necessary). Also, there are options like building underground and building multi-story highways.
* Infrastructure: The examples you're giving (power plants, waste disposal) are just discrete "points" in the map that don't have to be physically in any one particular place, as long as they're in the general area. Surely someone in a large area will be willing to sell you a plot of land. (If not, just build the infrastructure further out.)
Your point about density of USA vs. UK is a good observation, though I don't think it really changes the principles.
A lot of times American libertarians (I'm not one, FYI) talk about limited government, and their opponents immediately say things like "What about roads? Roads only work if the government builds them, and has eminent domain." I personally don't agree with the latter point of view on roads - but, in general, I don't think it's really an argument worth having. There are more important things to argue about, such as defending property rights in principle. But for a principle to be valid, it has to always be valid. So when I argue about eminent domain, I do so to save property rights, not because I acutally think eminent domain is particularly important compared to so many bigger issues in our culture today.
Heathrow is _hugely_ crowded; as I understand it has an activity level far higher than any other airport with two runways. They lobbied _very_ hard for a third, but it was ultimately shut down by a change of government. However, it wasn't really built in the right place (for the long term; it was fine then and for many years afterwards) and as a consequence there simply isn't anywhere to build another runway that doesn't require compulsory purchase of large areas - have a look at the maps if you're curious. One could say that they should have bought more land to give them more long-term capacity, but it did open in 1946 so I think a substantial degree of foresight would have been required! Its (constrained) runway length dates back to 1970, its core terminal location was designed on the premise that car parks were unnecessary because passengers would be being delivered by their chauffers. The world has moved on a lot.
As for building another airport - there's already (in rough descending order of size) Gatwick, Stanstead, Luton and London City also serving London, but none really has the critical mass to take over and all have separate constraints of their own, plus the road and rail infrastructure that currently links Heathrow with Central London would need duplicating on any of them. Actually, critial mass is the key issue here - Heathrow is a regular hub airport for a lot of routes. That simply wouldn't be an attractive proposition split over multiple sites.
Roads and Railways: We're currently seeing a lot of debate about the expansion of high-speed rail, very viable in Britain with our travel plans but much later adoption than the rest of Europe and the population density in Britain makes it very hard to find a route. The first line (to the Channel Tunnel) was bitterly opposed on every possible route and the subject of a lengthy fight, the second (initially London to Birmingham, then up the west coast) is faring little if any better. Look at a map; the line must be relatively straight if it's to work for high speed trains and must feed into existing terminal stations, but Britain is both crowded and far from flat. There are only so many routes....
Infrastructure - I suppose that is more of a general 'Not In My Back Yard' problem, a depressingly familiar attitude. Heathrow was cited beforel it's not long opened a fifth terminal (for two runways remember) and the planning enquiry for that had so much resistance that it took four years to complete. For a fifth terminal at an existing large airport that's 64 years old. Power stations (fossil fuel and nuclear requiring transport infrastructure for fuel supply and maintenance, renewable requiring large amounts of land), waste incinerators or dumps, high speed rail that by definition must go through areas it cannot serve or the station frequency would be too high - we all want their amenity in general, but somewhere else. The population density of England is too high to put everything 'somewhere else' though - we sometimes need powers to force development to ensure it happens anywhere.
Or for a fictional example. In a UK radio soap opera, there was recently a story where two rival property developers clashed. One set up a large scheme, obtained the sites, set up all the details - then discovered at the last minute that there was a previously unknown problem with the access to the site from the area they had been planning to use. Enter developer two who had bought a tiny parcel of land when he was what was starting, and which was now the only viable means of access to the site. Valueless land otherwise, yet he held the entire development otherwise to ransom. Now, in this sort of case compulsory purchase isn't available in the UK (and rightly so), yet he was in a position to ask for a vastly inflated sum because he became the keys to the whole project. Compulsory purchase stops that from being possible and requires all parties concerned to be fair with each other.
I wholeheartedly agree that forced sale should be a last resort and to be avoided wherever possible. If something is considered a vital piece of infrastructure though and there simply aren't available alternatives, I would rather have the option available to force a reasonable, fair and transparent deal through than to allow development to be held to ransom or flat vetoed by a single individual.
* Airports: How much is it really going to help to build one more runway? What about when another runway is needed, and another after that? The best thing to do is to plan for the airport to take up an enormous amount of space beforehand, and probably, in this case, to build another airport. (NYC already has three, but I only know of two in London off the top of my head.)
* Roads and railways: I think this is the "most difficult" one to deal with. I think it helps to have a more accurate definition of property rights than what most people operate with, which is to say, it's only your property if you actually use it--otherwise nobody is impacting you if they use it, which is the whole point of property. For example, I don't think people should be allowed to "own" large, empty tracts of land. (This would be the common problem with building roads and railways in America.) And if you want to build new roads in a heavily urbanized area, I think that's just a bad idea... I mean, that's never really necessary, unless you want to increase the human density of that particular area (which it self is never really necessary). Also, there are options like building underground and building multi-story highways.
* Infrastructure: The examples you're giving (power plants, waste disposal) are just discrete "points" in the map that don't have to be physically in any one particular place, as long as they're in the general area. Surely someone in a large area will be willing to sell you a plot of land. (If not, just build the infrastructure further out.)
Your point about density of USA vs. UK is a good observation, though I don't think it really changes the principles.
A lot of times American libertarians (I'm not one, FYI) talk about limited government, and their opponents immediately say things like "What about roads? Roads only work if the government builds them, and has eminent domain." I personally don't agree with the latter point of view on roads - but, in general, I don't think it's really an argument worth having. There are more important things to argue about, such as defending property rights in principle. But for a principle to be valid, it has to always be valid. So when I argue about eminent domain, I do so to save property rights, not because I acutally think eminent domain is particularly important compared to so many bigger issues in our culture today.