Filling in the gaps

I don't think you're going to be able to fill the gaps and get it to look right and stay stable. If you want a timber floor, I think you've going to have to replace the timbers.

Depending on the profile of the timbers you might be able to pull them up and nail them back down with the gaps closed up - of course you'd wind up with a space you'd need to fill with some new timber, but you might be able to hide that.

If the profile of the boards is wrong (edges heavily rounded over for instance), you might just be outta luck. If the boards are heavily warped, cupped or split you'd definitely be better to replace them.

Before you do anything significant, try sanding back part of one of the boards and wetting it to get an idea of how the timber would look sanded & finished - may be that since it was verandah decking it may not come up well or look the way you want.

Sure you don't want to just lay the carpet back down??

For all those guys out there who thought this was an impossible task, I have to inform you that IT IS DONE!
For those of you who are not afraid of hard work, and have the patience, this is the method I have used.
Firstly, talk to your wife..then get on the net and ask the companies that supply products for gap filling for advice. My inspiration came from Parfix who directed a company on the Gold Coast to advise, and although I didn't completely go with their suggestion, here is what I ended up doing...
Go to Bunnings and buy some flyscreen plastic spline (5mm and 4mm). Force this between the planks, press down with the tongue of a VJ offcut, this gives you 6mm depth, fill the gaps with a suitable coloured gapfiller, sand the surface and seal.
This works, I have left this in place on some trial planks for 2 months whilst renovating the room and no shifting has occurred.

I would have a look at marine deck caulking it seems to be a similar situation to some boat decks I have seen.
Planks with 6mm gaps caulked in black & sanded flat. a process that has been arround since Noah.
looks good and lasts with salt water exposure & strong flexing.

cheers