Albert and Vera Dick became good friends with Thomas Andrews (they were dining companions) and apparently the shipbuilder led them throughout the whole ship, including the lower classes. I have no doubt he showed them the bridge as well. Might be mistaken, though.
I do have an answer but it's just old plain and sence really.
If they did allow one passenger to enter the Bridge,it will never end.
Passenger would tell his mates and they'll tell there mates and so on.Therefore the Bridge will have endless of passengers entering the Bridge causing the officer's not to work 100% to there ability. Therefore causing much more mishaps and mistakes and saftey as gone completely
Whether or not tours could be arranged, I don't really know, but you can be reasonably sure it wouldn't have been with Mr. Andrews. As part of the Gaurantee Group, he would have been a very busy guy.