I'm sorry DLB to keep labouring the point, but I still think you need to be clearer about who should be responsible for sorting out your problem.
Trek has nothing to do with it. The shop is wrong to suggest they have any involvement with this what so ever.
You have a contract with JE James. They sold you the bike. They had to ensure that the bike they sold you was 'fit for purpose'. They alone are responsible for ensuring the bike you bought from them give reasonable service. Therefore your beef is with them and no one else. Any retorts they might make about Trek, or warranties, or other manufacturers are completely bogus. Your demand for satisfaction is against them.
So, assuming they sold you a reasonably priced bike which their sales pitch and literature implied was a fairly good machine, that was suitable for your intended purpose, and that you have only used it for that purpose, and maintained it within the guidelines they gave you, then they must stand behind that product.
Their only way out of this is that the bike was cheap crap and you knew that when you bought it, or that you weren't honest about your intended use, or that you have abused the bike in some way since owning it. So assuming none of that is true, their only other cop out would be if they can prove that 8 months is a reasonable life expectancy of a wheel axle at that price point. IMHO they would struggle to do that.
Buying a 'Trek' branded bike from a shop like JE James meant you had reasonable expectations from your purchase. So their choice is either fix it, or you'll have it fixed and make a claim to recover your costs against them. Just firmly inform them of their obligations under the Sale of Goods Act and reject all the fluff and bluster from these people.