You are in the right area by using the sale of goods act 1979/94. You are also correct in returning the goods to your supplier not the manufacturer (section 14), goods must be of satisfactory quality this defined in section 14 (2A), in short it is what a reasonable person would regard as satisfactory taking into account all circumstances including price. You may wish to look at Brown v Craiks (1970), if the goods are defective there is a breech of section 14 (2 & 3) if this is sufficient to claim a breech of contract rather than a breech of warranty will probably depend on the fault.
You may have discovered there is a lot of bluff in the law, but a mobile phone is a single use product and if it does not work you are entitled to have it replaced or your money refunded. I suggest you consider writing to the supplier, by recorded delivery if you wish, remind them that under the sale of goods act 1979/94 you are entitled to have your faulty goods replaced or your payment refunded and if they do not you will consider them to be in breech of contract and will take action. Give them a time limit to respond such as 30 days, which is reasonable and English law likes reasonableness.