Wood Pigeon eggs blown out of nest.

Hello, I'm looking for some advice.

A pair of wood pigeons have built a nest in a conifer in my garden, near my kitchen window so I've been able to keep an eye on them.

We've got high winds here and it's been really windy for the last few days so I was concerned the nest wouldn't last long.  This morning, about 6.30am I found 2 eggs on the ground under the tree and one of them is cracked.  The other one is ok so I've brought it inside to keep it warm.

My question is when the wind dies down a bit should I put the egg back in the nest?  I'm not even sure if the nest is still in place, so I need to check that first.  The reason I ask is I don't want to waste the pigeon's efforts if the egg is no good, I don't know how long it was on the ground but it was quite cold when I picked it up.

The mother pigeon is still hanging around so she hasn't given up hope yet.

Any advice would be appreciated, thanks in advance :-)