Interesting question, and at least from my point of view, possibly more complex to answer than you may realize ...

I suspect, but currently don't either have the time or mathematical rigor to fully model this from first principles, that a lantern will get you more light in a north-facing situation. This is for a combination of reasons:

1. Reflection / Transmission ratios depend on the angle of incidence; steeper (i.e. "head on") means more transmission. Since the panes of a lantern are more-inclined than a flat rooflight (even on a pitched roof), expect more light through.
2. For north-facing windows (i.e. permanently shadowed), it's _ambient_ light that counts - i.e. the more or less even brightness of the blue sky (or, less-bright evenly overcast sky ...). A roof lantern, with panes facing two or four different directions, is again "catching" more ambient light at steeper angles (transmitting it inside).
3. Ambient light (blue sky) is polarized to a strong degree, and reflection/transmission also depends on polarization (strongest: 90 degrees from the Sun, or - "generally northwards"). On some times of the day, ambient sky polarization may be such that a flat rooflight will reflect a larger proportion of light than at others. Again, the differently-oriented panes of a lantern would balance this out more.
4. If indeed your geometry is such that the roof opening would not be in the shadow all day long but receive some sunlight at some hours ... then a pitched-north-facing flat rooflight would be oriented disfavourably at such hours (away from the sun, so getting the light at shallow angles, reflecting much).

I'm not a technical consultant for this kind of advice (only have a physics background, which makes me a decent smart-*ss on such questions), and I can't say out of my head if a lantern would give you twice-as-much light in your situation, or only 10%-more-light. And then, the glazing (coated, and how ? double/triple glazing ? water-white glass or ordinary "greenish" float glass?) also plays a role. If you want it quantified, contact a professional. Right-to-light surveys commonly do such calculations (if for a slightly different reason ... how much light do I loose if that five-storey block of flat is being built on my doorstep ...).

Also ... an alternative, in some situations, may be an extension with a ridged roof (i.e. two opposite pitches), and then two flat rooflights (facing east/west). Large roof lanterns don't come cheap after all.

Thank you, v useful. I wasn't sure what expert to ask. Ideally would want to create a 3d visualisation with sun movement to compare but not sure who produces those. I will look into the consultants you mention. We are considering a gable roof for exactly the reason you suggested as it would have east and west orientated sides. I think they are more expensive to build but as you say, may even out given lantern costs.

Sketch up has the ability to novel sun positron