As per a Jeremy Crawford tweet:
Channel Divinity creates magical effects (stated in both the cleric and the paladin). [...]
From the PHB:
At 2nd level, you gain the ability to channel divine energy directly from your deity, using that energy to fuel magical effects.
So they would not work inside an antimagic field which states:
Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it.
There seems to be some confusion centred around whether the deity is creating the magical effect or whether it is simply suppling the power for the Cleric/Paladin to use to create the effect. Channel Divinity states that you use the divine energy to fuel a magical effect. To me this is very, very different from the deity creating the effect itself.
Lets looks at one example:
Sacred Weapon. As an action, you can imbue one weapon that you are holding with positive energy
So, to me and I don't think I'm stretching the interpretation here, the deity provides the energy for an effect that the PC creates. Hence the PC's effect will not take effect in a AMF.