As you may be aware, under the Consumer Rights Act liability rests with the dealer although some manufacturers (Dyson are very helpful I understand) will deal directly. However it’s not very helpful if they don’t actually have the spares.
Legally however you only have recourse to whoever you purchased it from.
You can buy batteries for these cheap on the Internet but the non-genuine batteries have been known to catch fire. It happened to a good friend of mine’s daughter.
If you are buying a battery online be careful because they are sometimes described as “Genuine Battery” (rather than Genuine Dyson).
For example, there are dozens listed on eBay (I have been where you are) but there are only a couple of genuine ones at about 65 quid.
You cannot do anything about the manufacturer not supplying you.
You may have a remedy under the Consumer Rights Act against the retailer (who may be the manufacturer) for failing to supply goods of satisfactory quality but a lot would depend on how long you had had the item.
Goods have to remain of satisfactory quality for up to 6 years but how long really depends on the nature of the goods because you would not expect a 10p ballpoint pen to remain of satisfactory quality for 6 years, but you would expect washing machine and probably a vacuum cleaner