I think if I were trying to explain this, I would not spend much time on the Coriolis Effect, I would concentrate on what is actually happening in the atmosphere.
Also I think people understand Centrifugal Force better the Coriolis Effect, so I would try to base my argument on that.
I would start with a parcel of air, e.g. an imaginary balloon.
Air may not weigh much, but gravity does act upon the balloon, and yet it does not fall.
This is an opportunity to explain that the air pressure pressing on the bottom of the balloon is greater than the pressure on the top, and the pressure difference provides a force which balances gravity.
Digressions into altimetry, beer bubbles, and why hot air rises are in order here.
Next I would talk about Centrifugal Force, conkers on strings, etc.
In particular, the CF gets stronger as the rotation gets faster.
Locating our imaginary balloon over London, ideally with a globe to spin, or at least a diagram, I would explain that CF acts on the balloon as well as gravity.
The balloon is not moving, so there must be yet another pressure difference balancing it.
It can also be noted that the CF acts Southwards and upwards.
Digressions about the atmosphere being thicker at the Equator are in order here.
If our balloon is in some "stationary" air, i.e. there is zero wind over London, its true rotational speed is due to the earth's rotation, and the consequent CF must exactly balance a pressure gradient.
Now we are set up to explain the "same latitude Coriolis Effect".
If our balloon is in some air which is moving Eastwards i.e. a West wind, its actual rotational speed is a little bigger than the average air for its latitude.
So the CF is a little bigger, and so this air parcel will move South and up compared to the average air around it, and the Earth below.
Conversely, an East wind will lower the CF, and the air parcel will move North and down.
There is not much scope for vertical air movement, and so the main apparent effect is for moving air to "turn right" in the Northern Hemisphere.
If you work through the maths on this, it produces the whole Coriolis Effect.
MATHS:
Briefly, the CF (more precisely, the centripetal acceleration) is:
V^2 / [ R cos(Lat) ] where:
Lat is the latitude, v is the surface's rotational speed at that latitude, and R is the earth's radius.
If the rotational speed 'v' is increased by a wind speed 'u', the difference is:
(v+u)^2 - v^2 = 2vu + u^2 (but u^2 is small enough to ignore).
So the increase in CF is: 2vu / [ R cos(Lat) ]
but v / [ R cos(Lat) ] is Rot, the Earth's rate of rotation,
so we have: 2 u Rot
Finally, noting that the component along the earth's surface is sin(Lat)
we get the full Coriolis Effect: 2 u Rot sin(Lat).
FOOTNOTE:
Incidentally, I believe the common "Easy Explanation" for air moving North only describes half the Coriolis effect.
Imagine you are in a balloon in some air moving North, clutching a DI.
You would see your track move increasingly East, according to the DI, because of the "wind shear" with latitude.
(Or conservation of angular momentum, if you prefer spinning ice skaters).
This is half the Coriolis Effect.
You would also see the Geographic North Pole move increasingly west according to the DI.
This is the other half of the Coriolis effect.