Paul Ebbens wrote:

So really it is not a privatised brandname at all, so that means First has
no true reason to get rid of it just because previous company was named so?

Quite so. And when I did a bit of detective work the last time this
question came up, I discovered that the trade mark "Thameslink" was
owned by the SRA (presumably now DfT Rail):

http://tinyurl.com/re2jr

so it has nothing whatsoever to do with past or present franchisees.