Yes, that's exactly how it works. Multiple/split tickets can count as a through journey with the associated protections under condition 14.2 of the National Rail Conditions of Travel. The main requirement is that you stop at the station where you're changing from one ticket to the next, and that you allow the minimum connection time (MCT) anywhere you change trains. The MCT is 5 mins at Newport, which is the default value that applies to most stations across the country.
The right to take the next available train in the event of disruption is generally constrained by any route/TOC restrictions shown on your ticket. So if your Newport to Weston-super-Mare ticket is shown as "GWR only", for example, then it is only valid on their services - not that there are any other alternative operators you'd plausibly want to use in this particular example.
It's only if sticking to this named operator/route would mean you'd expect to be delayed by more than an hour that you have additional rights. These include the right to be re-routed to your final destination at the earliest opportunity under Article 16 of the Passenger Rights and Obligations Regulation (PRO), which is something that the TOC at fault (i.e. which cancelled the train or caused you to miss your connection) has to arrange.
In practice, Article 16 rights are very difficult to vindicate because train companies don't train their staff at all on its existence, leaving you to either rely on any ticket acceptance the TOC advertises generally (usually only done if there's significant disruption) or that individual members of staff are willing to offer. Otherwise you have to buy a new ticket as required, and then contact the TOC afterwards to claim this back. This is obviously not a great process and TOCs are often difficult about reimbursing new tickets.
