Looks like it is used in jam making, according to this the fruits vary in toxicity depending how ripe they are, the more bitter, the more toxic. I don't think we're talking about dropping dead after one or two berries but I'd still seek medical advice if some got eaten.
I would get up a ladder and cut off the branches on your side, bag up and bin. Are you friendly with the neighbour? Legally you should offer the cut off branches to them as they are theirs but the space they are growing in is yours. I doubt they will want them back though.
Some close ups of the fruit would be good for ID. You can't take the risk with small children IMO.
We have hacked it back to the boundary fence and instigated a daily clearing regime to look out for the fallen berries each morning to minimise the risk of him eating them. thanks everyone!
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https://pfaf.org/user/plant.aspx?latinname=Prunus+laurocerasus
Some close ups of the fruit would be good for ID. You can't take the risk with small children IMO.