Royal Mail is fine for posting domestic within the UK.
The problem comes when you try to send important documents internationally, especially things like passports or immigration-related documents, because they can get caught up in customs on the way, and are sometimes delayed for weeks, or even don't arrive at all.
So, when sending domestic post, you can use Royal Mail, and if you'd prefer, pay for the service that tracks and records the delivery.
But when mailing internationally, I would use a courier like FedEX or UPS for important documents, as they have their own customs and so there's no worry about the mail getting delayed or lost in the government postal system.