Absolutely not, for two reasons.
1. Do we really want to destroy its punctuality like has happened with the silly and misguided "direct trains from everywhere to everywhere" thing that has ruined Thameslink, turning it into a super-Castlefield?
2. London is in the bottom right hand corner of England. Therefore the demand is skewed to the other side of London. The demand for stuff like Bristol to Kent is minimal and so this is not worth doing. It's the reason why there are arguments for a through HS2/NPR station at Manchester Piccadilly (though due to cost it is unlikely) and there is no real argument, outside of the UK joining Schengen*, for one in London.
* But even so, what would you do with about twenty five trains per hour approaching London from the north and west? There would never be demand to continue more than a tenth of that across to Kent and mainland Europe.