If it requires verbal or somatic components, no.
If wearing armor can be too distracting for the spell caster to properly focus and make the precise gestures needed (PHB 201) then I would think being bound and gagged in a way that prevents you from speaking or gesturing precisely will prevent you from casting spells with verbal and somatic components.
"If you can't provide one or more of the spell's components, you are unable to cast the spell....A character who is gagged or in an area of silence ... can't cast a spell with a verbal component.... If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures. " (PHB 203)
But the above rules apply to casting spells. Some Arcane Tradition abilities involve casting--in which case the above rules apply--and other do not.
Example:
School of Abjuration: Arcane Ward requires casting a spell, so this power just can't be used, but I would argue that Project Ward would work, unless the wizard was also blindfolded.
School of Conjuration: Minor Conjuration would work because you are not casting a spell, you just use your innate powers of conjuration to conjure something. What is great for this power is that even if blingfolded, you can conjure something in your hand. If you want to conjure something in unoccupied space, you need to see it, but not to conjure something into your hand--like a knife to cute the ropes that bind you! Benign Transposition states that you can use your bonus action to teleport 30' to an unoccupied space that you can see. It doesn't state that you cast teleportation. You powers of abjuration are so great that this is just an innate ability, like a blink dog. You are not casting a spell to teleport, you just do it. Again, a blindfold thwarts this.
School of Divination: Portent will work if bound, tied, and blindfolded and can even be used to replace an ability check used to break or contort your way out of the ropes that bind you. The Third Eye would work, but would be useless if blindfolded.
School of Enchantment: Hypnotic gaze would work, but you can thwart it by blindfolding the wizard. Instinctive charm can be used to prevent or end a torture session, but not if the wizard is blindfolded.
School of Evocation: There is no ability for this school that can be used when bound and gagged.
School of Illusion: Illusory self would be helpful to perhaps delay torture.
School of Necromancy: Inured to undeath is not affected by being bound or gagged. Not sure how useful it would be unless you are being tortured with necrotic damage. Command Undead will work if bound or gagged. But if gagged, it will become friendly, but you will not be able to give it any commands.
School of Transmutation: I would rule that minor alchemy's 10-minute "procedure" requires use of your hands at least and, therefor, cannot be done if bound. Master transmuter; if you have created a transmuter's stone and have it on your person, this ability would work and the major transmutation effect would help remove your bonds.
LESSONS LEARNED: OR HOW TO TO KEEP YOUR WIZARD IN CHAINS
(1) If you have to bind and gag a wizard, make sure to blindfold him or her as well!
(2) School of Conjuration is the best equipped for escaping from being bound, gagged, and blindfolded. If you capture a Wizard of the School of Conjuration, make sure to keep an eye on his hands. It is not enough to just tie them. At high levels, you will also need some powerful magic spells like forbiddance or magic items like dimensional shackles to keep a conjuration wizard bound.
(3) School of Transmutation is the second best equipped for escapism, but only at high levels. Starting at 14th level a school of transmutation wizard can turn his rope or chains into spaghetti. Don't tie your high-level transmutation wizard up with mundane rope, get a Ball and Chain +1.
