Hopefully since this topic is in the speculative discussion section this post fits the criteria. If you disagree with this post that's fine, but please can I ask anyone who reads what's below be polite and kind; I'm new here and a bit nervous to post.
Here goes...
When thinking of a livery/liveries for GBR, in my opinion, whoever decides it need to think about the following point:
How can the livery/liveries of GBR trains make the railways feel easier for passengers, no matter how often they travel, to use?
The priority of the railways and therefore of GBR is to offer passengers a service that is simple, reliable and inclusive for all to use. A service that people, no matter how often they travel by rail, feel confident to use and feel is hassle/stress-free. A service that even during disruption doesn't cause confusion or frustration. The creation of GBR is a chance for the railway industry to offer something that feels like a clear reset to the majority of passengers, as well as make them feel that the railway is changing for the better, regardless of whether it actually is or not. An example of this not related to this thread would be the removal of TOC-specific tickets.
For that reason, if I had god like powers and were in charge of GBR + the decision of a new GBR livery, I would make colour a big theme as this is one of the most simple ways a passenger can identify their train. From this, I would choose one of the following 2 options:
(Please note before reading, any colours listed below are just examples of colours for the purpose of the post and not a definitive thing. I'm not seriously suggesting a bright orange XC train, I'm just using orange as an example because it's a colour I like)
1. Have 3 or 4 GBR liveries that identify the train by the type of service that it offers. For example:
Intercity/High Speed Trains (Current examples include but not limited to LNER/EMR Intercity/Avanti/XC Voyagers) would be red.
Semi-Fast/Long Distance Trains (Current examples include but not limited to EMR Connect 360's+EMR Liverpool - Norwich/XC 170's/TPE Liverpool - Cleethorpes) would be blue.
Local Vast Majority or All Station Stops/Slow trains (e.g. Northern Leeds - Nottingham, Northern Man Picc - Sheffield, EMR Crewe - Newark) would be purple.
Very Short Distance Within A City Or Metro Services (e.g. London Overground, Merseyrail, any future Bee Network for Manchester) could retain their own local branding/colour.
2. Have 5/6 GBR liveries sorted by region so they don't overlap each other. You could repeat the colour if there's no or extremely limited chance of those services meeting each other and/or merge operators where possible. This would be trickier to do but something like (based off current operators):
SWR, Southeastern, Greater Anglia, LNWR = Blue.
Southern, GWR, C2C, Trans Pennine Express = Green.
LNER, Avanti, Chiltern = Red.
EMR, WMR, Great Northern = Purple.
XC = Orange.
Northern, Thameslink = Pink.
Why do this? So passengers can not only identify a service local to them but also a service type. Some examples are:
If stood on Peterborough station, a blue GBR train would identify the train heading east/south east. A red train would identify the fast train to London, Scotland or destinations to the north without clashing with open access operators. A purple train would identify the slower train to London or the slower train to Doncaster (I'd reclassify Liverpool - Norwich as XC). An orange train would identify a train crossing the country east - west.
If stood on Sheffield station, a green train would identify the Cleethorpes train. A purple train would identify the train to London. An orange train would identify a train crossing the country (remembering here I would classify Liverpool - Norwich as XC). A pink train would identify a slow all/vast majority of stations being stopped at service.
If stood on Birmingham New Street station, a white train would identify TFW. A blue train would identify a semi-fast train to Liverpool. A red train would identify a fast WCML train to London or Scotland. A purple train would identify a local or slow/vast majority of stops service. An orange train would identify a long distance cross country train. (I would classify the all stopping Birmingham - Leicester currently XC as WMR).
Obviously there are some examples above where colours would clash (LNER+Avanti at Glasgow or Edinburgh) but even here it simplifies the service for passengers who may not be familiar with travelling by rail. (Blue = Scotrail, Red = Fast Train to England/London, Green = Semi-Fast Train to Northern England, Orange = Fast Long Distance Train to England but not London.
In conclusion, I would make colour the focus of new GBR liveries to make the railway feel clearer and easier to use for passengers and also offer an easier description for staff, especially during times of disruption. The number one mission of GBR should be to create a rail network that people feel is simple, reliable, affordable and inclusive to use to therefore encourage them to use it as a regular choice of transport.