For some reason, whether you see the details first or not depends on where you choose a contact from. If you go to the Payments tab and choose a contact from there (either under “Frequent”, “Recent”, “All”, or by searching), it will take you to a screen showing the account details (sort code & account number). You then tap on the account details to go to the screen where you enter the amount.
If, however, you go to the Payments tab and then tap the “Pay” option at the top of the screen, any contact you choose from the “Pay” screen will go straight to the place where you enter the amount to transfer. You can try this with the exact same contact, once tapping on it directly from the main Payments tab, and once tapping on it from the “Pay” screen.
Finally, something else you can do (if you don’t merge contacts as @shizzle suggests) is to not use the payee’s proper name but put the account description in it. This will make things a lot easier. So instead of “Jane Smith”, set the payee name as “Jane - Personal” or “Jane - NatWest” or whatever best distinguishes that account.