It is true that we have done this before, here and elsewhere - but I like it each time
First I actually think it would have been much wiser to reconstruct Tiger than any of the OTL battleship reconstructions (Warspite, QE and Valiant). Of course any of those three wonderful ladies would throw a much heavier punch and be able to take much more punishment in a classical confrontation of the battle lines. The problem is however that by 20th century such battles were very unlikely to happen and in all of the OTL confrontations involving one of the three ladies a reborn Tiger would have done at least as well and in a lot of potential situations a Tiger with extra knots would have done a lot better.
IMHO the RN first of all needed ships that could actually engage the enemy as you couldn't count on the enemy necessarily wanting to engage (speed), and of course you needed ships that had a reasonable chance of being successful in an engagement (modernisation).
So instead of the three QEs I would send not only Tiger but also Lion and Princess Royal to reconstruction - and scrap three Rs.
You will still have plenty of old ships for convoy escort and shore bombardment, but you also have three extra ships with a reasonable chance of engaging enemy cruisers and "light" battleships (Twins, Italian reconstructions and Japanese Kongos) and provide escort for fast task forces.
I will not bother with reboring the 13,5" - it was a very fine and accurate gun with an adequate shell - reboring to 14" might risk reducing accuracy without getting a significantly better shell. The logistic advantage of sharing calibre with the new 14" is insignificant and you will probably need separate charges and certainly separate firing data. The guns will need min 30 degree elevation though and that requires substantial changes in the turrets.
28 knots probably would be possible well inside the existing machinery spaces and weight, but as the IJN succeeded in increasing the speed of the Kongos without deleting any turrets why not in the Tiger/Lions? 28 knot would IMHO be just enough, but I would seriously consider deleting a turret if you could get 30 knots and excellent protection.
Extra horizontal protection (min 4") and much improved AA armament is paramount however. The secondary guns will have to be DP, six or eight twin 4,5" would be fine plus of course some octuble pom-poms and later 20mm Oerlikons.
Aircraft facilities isn't important, but if you really feel like it put a catapult on one of the main turrets like in the Kongos.
And if the French would sell Strasburg and Dunkerque I would happily buy, they could have the some QEs or Rs instead - as many as they want!