Would it be possible for someone like royal mail or a large delivery company to develop and integrate blockchain into their logistics and payments system
Sure, but keep in mind that blockchain is, after all, only a database. And a pretty much restrictive one. If it's only one company using it, there are much more appropriate and easier implementations.
and then later be able to have perhaps their own tokens as a form of payment if they had their own blockchain that was a fork of say EOS or another network more suited to them.
Do you know those coffee machines where you have to buy tokens in order to be able to actually buy coffee? Did you enjoy that idea? I guess not. And I guess that your customers won't also like your tokens... unless they are not forced to use them (there's alternative still available) and they are more like incentivized to use your token (i.e. keep money locked in your system).
However, keep in mind that blockchain is not necessary for this setup.