Any UK based nappy shop that has been trading prior to 2004 I think it was when Paypal UK set up, is likely to have a merchant account and use a non-Paypal system for accepting card payments, although many do accept Paypal as well.
Look for services such as HSBC secure epayments, WorldPay, Protx amongst others, quite often the sites will refer to the service they use on the T&C pages..
Also you could look for sites that will accept payment by phone.
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