This possibly isn't what you're going to want to hear, in which case feel free to skip over, but I would say that you are extremely unlikely to be able to make a successful consumer law claim for this given the age and the fact it's a consumable part.
You don't get an automatic 6-year warranty with UK consumer law, it means you have the right to ask the retailer to consider a consumer law claim for up to 6 years. Consumer law is meant to cover inherent defects that were present in a product from day one, not to cover parts which have failed due to the course of normal wear and tear, and, let me be frank, bad luck.
The absolute worst case scenario is neither of you can reach an agreement (you want a free repair, Apple want to charge £199) and it goes to court. You would have to prove (at your own expense) that the product was inherently defective from day one, this would probably mean paying for an independent engineer to look at the product.
Other things that a court is entitled to consider is that you have had 3-years use of the product, and they are entitled to consider the use you have had out of the product. In summary, if they did find in your favour, it might be that a 50% repair cost be ordered, for example, rather than for a free repair.
It does seem unfortunate that the battery has failed after 3 years but there are so many variables with consumable parts including the charging and running habits, even things as minor as what temperature it is stored at. I would probably ask for a gesture of good will and say you're prepared to pay something as you understand it is consumable (perhaps they could meet you half way with the repair cost); you may not have much luck if you go in all guns blazing. If the enclosure has swollen Apple will usually cover any damaged parts (enclosure, trackpad, whatever), if you paid for the battery to be replaced. Ultimately £199 is quite expensive for a battery although not wildly outside what other computer manufacturers charge for modern built-in batteries, but you might be able to persuade Apple to meet you half way and charge £99 if you approached it in that way.
Like I say I'm sure you may not have wanted to hear this, but just my opinion really.
Many thanks.