(hint: PayPal is a wrong answer!)
I wish Stripe were serving customers outside of US, but this is not the case yet.
We recently had the same problem, mainly for recurrent billing, and the overall scenario is quite depressing in Europe.
The process of being accepted and setting up a payment gateway and merchant account is very long and require tons of documentation. We checked recurly, chargify and cheddargetter. All of them have some EU partners but they rates are often very high and responses are slow.
After two weeks of investigation right now I would suggest you cheddargetter (https://cheddargetter.com/). You'll still have to go through lot's of papers and bureucracy but at least their support (both from them and from their EU partners) is very quick. We set up an account, opened a ticket to be helped setting up merchant account and within 3 hours we've been helped and immidiately contacted by their european partner to set up our merchant account.
However, if you're not looking for recurrent billing, there could be more convenient local solution in your country provided directly by banks. Another respected solution in Europe is Ogone. We used them for a couple of our clients and they're at least decent :)
I don't know your reasons for not wanting to use PayPal, they might differ from my hesitations, but after spending a couple of days experimenting, reading docs and so forth, I found the technical side of it extraordinaraily flexible and easy to use.
- Barclaycard for your merchant bank account
- SagePay for the payment gateway
And if you're implementing recurring payments then I'd recommend Recurly.
After much trial and error this is the best combo that I could find.
If you're after something even simpler, try using hosted payment page solution from SagePay.
Make sure you take into account VAT when taking payments in the UK.
We settled on WorldPay, who provide the merchant account and the gateway. They are pretty terrible, but they get the job done.
Alternately, PaymentExpress (http://www.paymentexpress.com/) seemed like a great option.
i.e. authorize.net.
You don't need some gimmicky service to process transactions.