In principle yes, you should be conveyed by taxi, or potentially even minibus or coach if there's enough passengers. It's not unknown though unfortunately for TOCs to argue that they have no obligation to do this, especially with split tickets (of course the right to onward transport exists even with split tickets, the same as through tickets). As such, I'd make sure you had enough funds to pay for onward transport in case the TOC claims they don't have to pay for it, and be prepared to get in contact with Customer Services after the fact to claim back any funds owed.
You must stick to the minimum connection time (MCT) at each station on your journey - MCTs can be found at brtimes.com. In this case Carmarthen has a MCT of five minutes, with your itinerary allowing twelve minutes so you're covered. Also as a side note, the Carmarthen train leaves Piccadilly at 15:31, just so you can avoid any excess waiting at Piccadilly