The Swedish equivalent to Paym, called Swish, has been really successful. Probably an interesting case to study for those trying to create the next Paypal.
I doubt we'll see a new Paypal until there is a new marketplace (Ebay) on a new communications medium (internet) that attracts enough users that a new solution needs to be created to deal with it.
[1] http://ecommercenews.eu/swedish-banks-want-use-swish-ecommer...
Similar example: Polish mobile payments company (Blue Media) wanted to speed up most commonly used service - charging prepaid GSM accounts so they created account in pretty much every existing bank as internal transfers are pretty much instant. This also required to integrate with many APIs to fill out/automate transfer details. Thanks to that it was easy for them to offered express transfers (5-15 minutes instead of standard 2-4 hours)
In the United Kingdom, sending bank to bank or Paym is all the same if I'm not mistaken. The only difference is that on Paym you'll use a phone number instead of an account number.
"Paym is a new service launched by VocaLink in 2014"
"This money goes to Vocalink, a company owned by a consortium of 18 banks, which operates FPS, along with Bacs (for Direct Debit), and the LINK network (ATMs)."
I would say it's likelier that the fact that Swish transactions are instant from the customers perspective have played a large part in why the uptake for Swish has been so good in Sweden. As regular bank transfers still take hours to days to settle.
Yet the British banks have done very little marketing. I wonder if they prefer the current situation, where they perhaps earn more money from debit card fees.
If you offer a faster and cheaper way to settle payments than your competitors, you'll be able to gain a lot of lucrative corporate customers.
However, if all of the major banks offer a faster and cheaper way to settle payments, then they all simply earn less in fees and also have to pay for developing a new system. This is a loss-loss situation, so it's going to happen only in face of strong competition by other payment systems (in which case they can near-immediately lower prices and increase speed to undercut them, while still making profit) or by regulatory means - e.g. the SEPA-payment legislation in EU forced many inter-EU payments to become much faster and cheaper than before.