My wife and I booked two tickets via Trainsplit from Neath to Brighton via Paddington and Victoria. The barriers at Paddington Circle Line would not let us through.
It is not uncommon that tickets don't work barriers; this can be for a variety of reasons (e.g. improper encoding by the ticket machine that dispensed the tickets, or the magstripe was damaged, which can happen if it gets near anything magnetic, etc...) but staff
should let you through, on presentation of the tickets.
GWR said it was TfLs problem, Ticketsplit refused to accept the tickets wouldn’t work and blamed TfL.
They are both correct, TfL/LU are to blame for this. If their staff refused to let you through, the best thing to do would be to tap in and out, and submit a refund request to TfL. Unfortunately some TfL gateline staff do refuse to allow passengers to travel with valid tickets; this is a known problem, which TfL doesn't do enough to rectify.
We were unable to complete our journey until we booked a taxi from Paddington to Victoria at a cost of £25.# GWR said our tickets didn’t show the tube journey. So we were late for our next journey and £25 out of pocket with no one owning up.
If you missed the train from Victoria, as part of your journey from Neath to Brighton, this is likely due to taking a taxi instead of taking the Underground. It's 14 min by Underground (excluding time to get to/from platforms, wait for a train) but at busy times would be 35 min by car (excluding time to book the taxi and get to/from the pickup/set down locations).
TfL may be reluctant to refund you for a taxi, as the appropriate course of action would be to tap in using a Contactless card. If you do not have a Contactless payment card, there are ticket machines available which take cash and cards, albeit at a higher fare, but still much less than the taxi fare.
I don't understand how you can say that "GWR said our tickets didn't show the tube journey" when you also said that "GWR said it was TfLs problem"; perhaps different people at GWR said different things? This is not uncommon for GWR, who have poor standards of staff training at stations such as Paddington, and poor standards in their customer services department too.
In summary, the dispute is with TfL; TfL are liable and must pay compensation, however TfL may be reluctant to pay more than the cost of the cash fare from Paddington to Victoria.