I have NO idea how things work in the UK, but here in the US, there are special (usually less formal) rules governing small claims that make them more friendly to consumers claiming an action against a business.
If you had a contract that guaranteed delivery, then that's it. Game over. The company owes you money. But I'm guessing you have so such contract, which is why you're posting here :-)
This might be useless information in the UK, but a small claims case brought against the company in the US would likely center around whether or not your agreement constitutes an implied warranty of service. To win, you might have to show negligence or fraud on their part (this is where the different rules would apply -- the bar is lower for small claims). The judge would have to consider the following in the absence of any writing, among other things:
- How much did you stress the time sensitivity of the delivery?
- Was the failure to deliver on the agreed-upon date within the company's control?
- Was the company aware of the cost of late delivery to you?
- How much did their promise influence your decision to do business with them? In other words, did they fear that they might lose the sale if you thought they couldn't deliver on time, and how hard did they try to overcome that worry? Or was it not really discussed that much?
- Were you being reasonable in your expectations (i.e. are delivery dates commonly missed in that area/industry, or do things usually go as planned?)
I successfully argued a similar case in small claims court and won on these grounds. I had purchased a king-size mattress for an upstairs master bedroom. The store's policy was that I could return the mattress within 90 days for any reason, but I would be charged a 25% restocking fee if I did. I paid extra for delivery and setup of the new mattress along with haul-away of the old one. I made it clear multiple times that there was a 90-degree turn involved with getting a king-sized mattress up the stairs. I informed the salesman that I wouldn't buy the mattress if they weren't able to get it up there, and he assured me that his would be no problem at all for them.
Two weeks later my mattress arrived (they had to order it), and sure enough, the delivery people were not able to get the mattress up the stairs. They offered me a choice; if they bent the mattress to the degree needed to get around the turn, it might get damaged and they would not honor the warranty if that happened. Or, they could take the mattress back and I would be charged the 25% restocking fee. I chose neither, and it even resulted in an argument with the store owner at my house. They took the mattress back but would not refund the restocking fee. I didn't like suing, but I was on the hook for an almost $400 fee for a mattress that never even came out of the plastic wrap.
I won the case. The judge ruled that the store had every resource to know the logistical challenges with getting that mattress up the stairs because they also owned the moving company and have delivered to similar houses in my neighborhood before. He also ruled that I demonstrated due diligence in making them aware of those challenges and that I made it sufficiently clear that I would not have made the purchase if I believed I'd face a $400 fee if the mattress didn't fit (they argued that I took the risk anyway in spite of the fee).
Basically the judge ruled that the actions the store took to address my concerns in order to make a sale is evidence that an implicit guarantee was made, and that constitutes a warrantable claim on my part.
If you can show that you made your concerns about the delivery date clear to the company -- that it would cost you dearly if they didn't deliver on time -- AND that they overcame those objections by making you a promise that they then failed to uphold, that might be sufficient to win a judgement against them. But again... U.S. law vs. U.K.