My goal is to avoid a confrontation with the ticket office stuff. Leamington and Warwick are staffed, so I suppose one of those should work. I managed to get the forum ticketing website to offer me a gold fare for Kenilworth - Oxford off-peak return by requesting a change in Banbury. Assuming I have such a ticket: Kenilworth - Leamington Spa (WMR), Leamington Spa - Banbury (Chiltern), Banbury - Oxford (XC), the bottleneck seems to be LMS - BAN, which is restricted to Chiltern only. Just to confirm, is it correct that if the intended Chiltern service is delayed/cancelled, I would be stuck at LMS for an hour waiting for the next one?
If the expected delay in reaching your final destination would be 61+ minutes by sticking to the terms of your ticket (i.e. effectively Chiltern only between LMS and BAN) then you would be entitled to exercise your right to "re-routing at the earliest opportunity" under Article 16 of the Passenger Rights and Obligations Regulation.
This right would need to be exercised against the operator at 'fault' for your delay, i.e. WMT if you miss your connection at LMS due to disruption on the Kenilworth to LMS leg, or Chiltern if it's due to problems with the LMS to BAN leg. As per the other thread, generally I would suggest PRO rights are best exercised in writing (e.g. via social media) so that there is clear written evidence of you having exercised it.
The right to re-routng would, in practice, require WMT/Chiltern to either endorse your ticket to be valid on another operator (e.g. by giving you a stamped form), giving you written permission, issuing you a zero fare excess, or buying you a new ticket.
Of course we know that in practice all but the first option are vanishingly unlikely to be provided, and since Chiltern manage LMS, their station staff may well refuse to do anything on behalf of WMT if it is their train that has caused the disruption. (I can't say that specifically for the LMS station staff, but it has been my experience at other stations served by multiple operators.)
In the event the operator at fault refuses to honour your re-routing request, I would suggest buying a new ticket as required, and then submitting a complaint to customer services after your journey asking for reimbursement of the cost of that ticket. Unfortunately when I've done this in the past with respect to other PRO rights, it has often required me to send a Letter Before Claim or even commence legal action before the TOC has agreed to pay up.
It shouldn't be this complicated, but unfortunately it often is. Once Chiltern and XC come into DfT ownership, the situation will improve slightly as there is automatic ticket acceptance between DfT TOCs in the event of cancellations.
This generally applies to the 2 trains before or after the cancelled service, or in some cases this is extended to 2 hours either side. For inexplicable reasons this acceptance doesn't cover missed connections or delays, so there are still gaps where you would need to rely on your PRO rights.