Unless there is a new bug that prevents you from going to the TOS Bridge, then it should be available just like any other bridge. Of course, Nukara and the Fleet Starbase prevent you from visiting the bridge so that might be the problem.
There's a little pull down menu on the bottom right corner of your minimap.
Click that and click visit bridge.
I was once DKnight1000, apparently I had taken my own name so now I'm DKnight0001. If I ask you a question it is not an insult but a genuine attempt to understand why. When I insult you I won't be discreet about it, I will be precise and to the point stupid.
It is, in my opinion, the best of the bridge and interior sets, but it does have some issues.
First, the new Duty Officer contacts are not in TOS uniforms, making them totally out of place.
Second, there are several NPCs using tricorders that continuously squeal the TNG tricorder sound (not that the TOS one would be any better). This is an issue I have throughout the game, it's made me turn off my sound more than once, as the unending squealing sound gives me a headache. It's made the Engineering section unusable to me.
Third, there are a couple of issues with the actual modelling.
One, the main bridge is lopsided. Look at the consoles to starboard, and then to port. They're at different heights.
Two, everything is STILL too big, though it's better than most other interiors. The helm is like a football field. Watch Balance of Terror. Kirk walks past and it stands about hip height. Here, it's shoulder height. This issue appears throughout the ship.
Three. The corridor to Sick Bay ends at the Sick Bay door, when it should continue, at least to the science lab (with doors to the medical office and lab).
Four, The Engine Room is the one from the first season, which is all very nice, but the later version of the room is better known, and more interesting.
The Transporter Room is wrong. The actual pad is turned clockwise too far, and the monitor station to the left of the console is absent (see the Changeling for what I'm talking about). Of course, the pads are too big as well, but that's been covered.
Don't forget to check out the Jeffries Tubes between decks! Access them by the ladders on the crew and engineering decks.