Do they apply to purchases of services as well as of goods?
For such purposes, what constitutes an online purchase? Must it be automatically accepted card payment, or does a discussion with a service provider and their provision of their bank details to you transfer?
How long does one have to change their mind?
What are the relevant statutes?
The purpose of the online refund right is that buying online, you haven't been able to check out the goods in real life. You buy a t-shirt. In the store, you would notice that it's colour is awful, from a screen shot in an online store you don't. So the right just puts you into the same position that you would be in in a brick-and-mortar store.
The critical element isn't how the purchase was made, but whether you had a chance to inspect the goods for real, or not. According to https://www.gov.uk/accepting-returns-and-giving-refunds the store has 14 days to offer you a refund if you tell them you want to cancel, and then you have 14 days to actually return the goods, and they have then 14 days to refund the money. All starts at the point where you receive the goods.
In the UK the rules depend on the good or service in question, the price and whether the good or service has been downloaded, used, opened or such.
Assuming the seller has provided no information about which law applies to your purchase (your contract) - which depending on the circumstances may be an offence - perhaps first look at The Consumer Contracts (Information, Cancellation and Additional Charges) Regulations 2013 and "the limits of application" in sections 6, 27 and 28.
If the Consumer Contracts regs don't apply to the good or service your contract is about, don't take it as read that there is no right to cancel or cooling off period or such - another rule may apply. For example there is a 14 day cooling off period for insurance contracts made online and you could consult the FCA handbook for information about that rule.
Which? magazine and the UK's citizen's advice organisations offer good guidance about UK consumer rights. It seems infeasible to give a comprehensive answer here about all goods and services and circumstances.
