Comment by kragen
Basically this happens because the DVLA and the stock market don't have any competition. Customers in a competitive market won't be angry when you need to be offline for 12 hours every Sunday morning; they'll just switch to your competitor some Sunday, because the competitor is providing them something they value that you don't provide.
The stock markets definitely have competition. For instance Frankfurt, London, Paris or Amsterdam very much compete with each other to offer desirable conditions for investors, and companies will move their trading from one to another if it is their interest. I think the fact they close at night is a self-preservation mechanism, traders would become insane if they had to worry about their positions 24/7.