Forgive me but there is some absolute twaddle being talked on this thread.
A battery is a consumable. So is fuel. Have you found that the car tends to run out of fuel from time to time? When it does that, does your opinion about whether the car is fit for purpose change? No. It remains entirely fit for purpose and as described. You should always be very wary of people expressing absolute opinions about legal matters - the law is rarely that clear - but I should be astonished if you have any right to reject this vehicle.
Consumables will not be included in a warranty, nor would it be assumed that they were in the event that a warranty did not mention them. Rather any court would be likely to want to see a specific term in the warranty to say that the warranty covered tyres, fuel, windscreen fluid, wipers, batteries and the other things that one gets through using a car, if it was going to find that they were covered.
You could spend £80 on issuing a claim in the small claims court, have a directions appointment and a trial (so requiring two attendances at court for which you will not get any recompense) and in about April of next year you will get a result. You are likely to lose as you only have partial evidence. It is in the nature of modern batteries that their performance can fall off a cliff. They work. And then they don't. The model we remember from 30 years ago of a battery getting weaker over time just doesn't apply any more. There is a long explanation why but this post isn't the place for it.
So consider this possibility: that the former owner drove it on the motorway every day thereby charging it, so even though it was weak (which would not be unusual with a ten year old battery) that fact was never noticed. It was taken into the garage that sold it and sat about for a couple of days, maybe including a couple of battery-draining short journeys. It started a few times but then came the time it didn't. Perhaps the lad who went to fetch it couldn't start it, hitched up the booster that ALL reputable dealers have on their forecourts and brought it round, running, to the salesman. Did the salesman know? If he did, would he have thought it was sinister given that this is a daily phenomenon when selling old cars? Can you prove that he misled you? If so was it a misrepresentation such that you can rely upon it to repudiate the contract? Law really isn't as straightforward as some will make it out to be. You don't know enough about the history of this car or even what happened in the few days prior to buying it, nor are you in a position to get it. The cost in time and perhaps money in trying to gather that evidence will make the price of a new battery pale into in insignificance.
If I were the garage I would certainly defend the claim as I'd want to avoid the precedent of someone having extended a warranty in this way. I might send a member of staff so as to be able to claim half a day's loss of earnings from you.
BUT! Once we have calmed down and stepped away from the law, it is a bit shitty to sell a car with a dodgy battery. Trading standards will not have the slightest interest as no law has been broken, but the garage might well be concerned to sustain its reputation for fair prices and fair dealing (if it has one). If you in an entirely reasonable and friendly manner were to contact them, point out how disappointed you are that they have sold you a car with a dud battery, acknowledge that it is not covered by the warranty but point out that one might reasonably expect a garage of their repute to replace it as a goodwill gesture - without any admission on their part that this was a knowing error on their part - they might well do so (the trade price for the battery being rather less than you would pay). Again, if I were them I would do so and probably say sorry while I did it. If you threatened me with law or trading standards I'd invite you to get on with it.
Good luck with them. They have no legal duty to replace it, but I for one think their moral duty should extend to a battery!
HP