Southeastern rebranded under state control, and then again 2 years later when franchised.
Don't forget guys that the Government owns the rights to the names of the shadow franchises used prior to privatisation. Some private companies continued to use these names to market their trains (eg: Stagecoach reverted to South West Trains after a short period of Stagecoach SWT; Gatwick Express; ScotRail; Thameslink until FCC took over; WAGN; Southern dropped the Network but kept the South Central post Connex and prior to the full launch of the Southern brand etc)
I think the East Coast Mainline was marketed as East Coast with a rising sun or iron wheel logo, prior to privatisation and GNER. It could become East Coast again with a simple logo or typeface. If stock needs urgent repainting then I guess it's the choice between white without the National Express lines, or GNER blue.
The pre-privatisation SouthEastern, became Connex South Eastern at privatisation. The post-Connex, government-run SouthEastern obviously has to drop the word Connex, and came up with a simple logo and typeface.
The logos were only used on the trains, posters, timetables and websites but I don't believe on any permanent features such as signs. Any mention of Connex at stations was taped over. The government-run SouthEastern retained and continued repainting units in Connex's white/grey/yellow livery (rather than adopt their own livery) but repainted stations in basic dark blue.
GoVia continued with the marketing name SouthEastern, although I think the company name had to be switched from SouthEastern Trains to SouthEastern Railway. While the logo was changed, I think they are to be applauded for continuing to use the neutral white/grey/yellow livery first introduced by Connex on their inherited stock, even if they have tied with the odd idea at a repaint. (Blue trim on 508s, and odd Electrostar.)
I think any incoming franchisee has the right to use the Government-owned names, so we could one day see the return of Thameslink, plain but simple Great Western and Great Eastern, or the dull LTS Rail and Network SouthCentral.