It can talk

There's a difference between

The imp can use its action to polymorph into a beast form [...]

And:

The imp can use its action to cast the spell Polymorph on itself to morph into a beast form [...]

The imp is not casting the spell Polymorph, it's simply changing its form into a beast. Since the description doesn't say it loses the ability to speak, there is no rule stopping it from talking.

Yes

The ability is not the Polymorph spell

The weretiger cannot possibly be casting polymorph, because polymorph requires that the target change into a beast. However, the hybrid form is distinctly a humanoid form. We can deduce this because the ability says:

Its Statistics, other than its size, are the same in each form.

Monster statistics include type. Which means that even in tiger form, the weretiger is a Humanoid (Human, shapechanger).

Polymorph says that:

The target's game Statistics, including mental Ability Scores, are replaced by the Statistics of the chosen beast.

If the weretiger were casting Polymorph, its type would change. But it doesn't, so the creature isn't casting Polymorph. Therefore the rules for the Polymorph spell do not apply.

They can use weapons, which is probably good enough for somatic components.

The actions of the weretiger allow for the hybrid form to use weapons, with no penalty compared to their normal humanoid form. If their hybrid hands can wield weapons effectively, then there is no reason that they can't manipulate material and somatic spell components with similar ease.

Hybrid forms can still speak

The stat block for the weretiger allows the creature to speak in all forms except tiger form. Thus, the hybrid form can provide verbal spell components.

So, we have established that the limitations of the polymorph spell do not apply, the hybrid form can use weapons and so should have the dexterity for somatic and material spell components, and they can still speak to provide verbal components. Thus, in hybrid form, a weretiger can cast spells.

All other werecreature hybrid forms follow similar structure as the weretiger to my knowledge, and so should work identically.