If it's present-day, then this likely isn't a question for this site, but just a question of fact, best answered by going to primary sources. This is the kinda thing that, as authors, we should do ALL THE TIME, just like reporters.
The obvious thing is to contact the people who will know, and manufacture the TBMs. Click the little envelope icon on the top right of Herrenknecht's homepage, and say something like "Hi, I'm writing a story about some fictional labs that have been dug out using your machines. I want to make it realistic, so could you tell me the tightest turn radius of the m-1070m EPB4000ah model?" (I picked that one because it has 4.6m excavation, 3.6m finished tunnel radius, but go for whatever model you think best).
If your tale is set any distance in the future, though, I see no worldbuilding reason that the engineering challenges associated with drilling any angle you want for your narrative could not be assumed to have been resolved. Still worth contacting Herrenknecht to establish a max value for the angle, then you can reduce it as you feel appropriate for your story.
(I'm no TBM engineer, nor do I have any special knowledge in this area, so take what follows with a pinch of salt)
If you're cutting curves, the limit seems to be a factor of:
- how long your segments are, plus
- the maximum angle between each segment.
Another limit is:
- the maximum angle that the cutting head can be turned to, and still be pressed against the cutting face, and still have the area between it and the TBM be fully braced.
The constraints on these are not necessarily entirely engineering limits, since smaller-diameter TBMs can be created, with shorter segment lengths.
But SOME engineering limits exist. Let's say that you're lining the tunnel in 4.5m-long sections of reinforced-concrete liner, 30 cm thick. Each segment needs (as a vague back-of-the-envelope guess) about 100 metric tons of liner.
If you're creating that liner in slabs of a quarter circle, 4.5m x 7.5m x 0.30m, that's about 25 metric tons of concrete, and there'll definitely be engineering limits on the minimum size of the equipment that passes that up the TBM, and places it against the wall, while also leaving room for other TBM requirements to be passed through it like a conveyor for disposal of waste material, coolant, lubricant, humans, replacement teeth and other repair parts and consumables, air flow, etc.
But there's no reason to have the concrete that large, other than to save time. You can make the liner in smaller parts, by reducing the sizes of the concrete sections. At an extreme, you could line the tunnels with bricks (and there're plenty of large brick-lined tunnels out there in the real world: brick performs excellently under compression!). It becomes a time-and-complexity tradeoff.
A curve in the tunnel would require shaving the respective angle off the bricks/blocks and tossing the dust and offcuts onto the conveyor of waste material, or having them be cast in precisely the right dimensions well in advance, and passed in for placement in precisely the correct sequence (which is how I believe it was done for the Channel Tunnel's large concrete sections, or so I vaguely remember hearing on some TV program about it).
In an ideal world, and for your maximum narrative freedom, you'd have the TBM-segment length be less than your tunnel width, the max angle between each TBM-segment be 90, and be able to have the cutting head be able to bore at 90 degrees to the main body.
At that point, you can bore whatever tunnels you wish.
The "making a 90 degree T-junction" thing could be a bit of an exercise: cut the crosspiece of the T past the junction, then back up leaving the main boring head in place against the cutting face at the far end of the T, so that smaller boring machines can be passed through the body of the main TBM, to expand the junction enough for the cutting head to then be pulled back and rotated to 90 degrees, or whatever angle required.
Keeping everything braced while doing all that, then placing the liner, might be the most significant engineering challenges there. That could be reduced if your site was specifically selected to be self-supporting rock with no sections of muddy, watery soil. But if you're looking at the EPB (Earth Pressure Balance) TBMs, then you're probably talking about digging through soft, cohesive soils.
