The first thing to do is to request service from Apple or an Apple Authorized service provider. They will diagnose exactly what's wrong, whether they can cover it and what the cost would be for an "Apple" repair, which for a home button means the device is swapped with a new/refurbished unit.
If they deny warranty service, you can clarify if it's due to time or some other reason. At that point, if it's purely a time issue - you might be able to request a consumer law exception here:
Just because Apple's warranty is one year, you are also covered any statutory time periods which in Europe is generally 2 years. It's important to present the device for service (online or in person) as soon as you can. Here are some pertinent footnotes from the page above:
In most EU member states, consumers may only claim for defects that were present on delivery.
The burden to prove that the defect (including latent defects) existed on delivery generally shifts to the consumer after the expiry of a period of 6 months from date of delivery.
In most EU member states, consumers may only claim consumer law rights against the seller from whom they purchased the product.
So if you didn't buy it from Apple, your consumer law claim would need to go to your vendor and not the manufacturer (since in that case they wouldn't be the same). Also, it's not an open-shut case since there could have been a latent defect present at manufacture, but it could also be a case where a physical switch failed due to normal use. I would presume Apple would take care of you if it were clear to them that there were a defect at assembly, but I don't know how they would procedurally come to that conclusion. I don't suppose each genius bar would open and look for a defect under the home button or if they would have a policy or if each case is handled as they arrive.
It's pretty easy to read up on this, but I would be interested in hearing how this actually went from someone in the UK to give you concrete advice.