The only thing I can see in regards to a network re-arrangement/improvement, without investment from yourself, is the Universal Service Obligation (USO) which ensures affordable access to the internet with speeds of at least 10 Mbps. Of course, this USO aims to ensure everybody at least 10 Mbps by 2020, so it is still some time away before you can anticipate it being applicable to yourself. Even so, Openreach are still unlikely to make network arrangements (because they are expensive, as j0hn said) even to meet the USO, but rather employ LR-VDSL (which simply stands for Long Range VDSL) to improve speeds beyond 2KM, with the exception of the aforementioned “technology” not being able to provide the level of service (in your case it may or may not, but we can only see once and if it is rolled out to you).
Alternatives, unfortunately, tend to be expensive because they are either niche or high maintenance, or if it is not subsidised then it may prove to be costly for the end user.
I would not recommend satellite for any gaming or latency-sensitive applications, or if you use a lot of data. I should put a disclaimer that I have never knowingly used satellite broadband myself, but I personally would certainly steer clear.
4G can be expensive, but then you have to use it reasonably and pick the correct package that works for you. What I can say for mobile broadband is that has become a lot cheaper than 4-5 years ago, and higher usage allowances are available. Three mobile, for example, has 40GB packages available from £21/mo on a 24 month contract (with shorter contracts available cheaper) that include Go Binge that allow streaming from services such as Netflix, tvplayer, etc. without using your data allowance. You could probably find a better deal, but I will leave that up for you if you were interested.
You say 4G was a bit hit or miss, but if you were to use it as your home connection it would be recommend to set up an external antenna and use a proper 4G router (and not some “MiFi” or dongle). There is a guy who uses the alias “DTMark” on ISPReview UK who used to post about 4G broadband. He would probably be one of the best people to ask about setting up 4G home broadband if you could find him.
It does say FTTPoD (Fibre To The Premises on Demand) is available in your area, but I would rule that out as being a choice if you are not looking to spend £1000’s in construction charges since, even with the new pricing, it is likely to be expensive as your nearest NGA Aggregation Node could be a few kilometres away at least. Then again, it is subject to a survey and there are providers such as Cerberus and FluidOne who would be happy to quote you, but if you did not proceed with the order then they will charge you the survey fees. Perhaps, if you were interested, you could even enquire with Openreach about where the nearest NGA Aggregation Node is just to give you sort of an idea of how far they would have to do works, but they don’t usually reply directly to consumers.
I would say look at the current voucher scheme’s terms (if one exists currently) and see what it exactly covers if you have not done so already.