There should be no tax consequences if everything is moved "in-kind". The basis info will only be a problem if it is missing when you sell. If you sell to cash before the move that would trigger taxes. If you have a loss, or are going to sell soon anyway, selling before the move may be easier. If the transaction charges at RBS are higher than Fidelity you might want to wait and sell at Fidelity.
We moved my mom from Morgan Stanley to Fidelity. We actually wanted to sell everything before the move, since we were also moving from individual stocks to index funds and Mom was not charged transaction fees at MS. However, MS did not sell on time and Fidelity pulled everything, not just cash. We ended up with a bunch of stocks to sell at Fidelity. All basis info came over just fine, Fidelity gave her 100 free trades, and I spent a day selling off a wild assortment of stocks. Kind of a hassle, but no basis problems.