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The 28 days is an internal policy. You may bring a claim against a company for breach of contract/ negligence at any point within 6 years.
You can issue a Small Claims case against them if they refuse to pay you for your loss.
It is always best practice to send a formal letter before action before you issue a claim. This is a formal letter whereby you tell the other side that if they don't pay you within 14 days the money they owe you, you will issue a claim. You can find some help on completing a letter before action here: https://www.which.co.uk/consumer-rights/letter/letter-before-small-claims-court-claim. I'd like to think that just this letter might "kick them into action". You'll probably find that you will get some resolution within the 14 days meaning you don't have to go to the extent of issuing a claim.
You can start a Small Claim by just filling in an online form. You can find this here: https://www.gov.uk/make-court-claim-for-money.
In terms of the fees involved, this depends on the sum you are claiming, you can find a list here: https://www.gov.uk/make-court-claim-for-money/court-fees. If it goes all the way to trial then there is also a hearing fee to pay which is a similar amount.
Assuming you "win" the case, you can claim back those fees from the other side though.
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