If your partner was living with the ex at the addresses concerned, then he is jointly & severally liable with her for the council tax, whether his name was on the bills or not.
Bailiffs are only allowed to take goods belonging to the debtor, so they should not take anything of yours or your son. However, if bailiffs do get in it can be difficult to prove to them that the goods are not your partner's - sometimes they can be quite difficult about this. You do not have to let them in. If they can get in through an open or unlocked door or window they can enter & take "walking possession" of goods, so make sure you keep all doors & windows locked. If they do come just refuse entry to them. They can take any goods left outside or in an unlocked garage or shed etc., including cars belonging to the debtor. If you have a car which is yours make sure you have the documents available to prove your ownership. This is not just the logbook (that is only a record of the keeper) but the invoice for purchase of the car. If your partner owns a car either keep it in a locked garage or park it well away from the house.
Your partner needs to sort out his liability with the Council & set up a payment arrangement. If he has not previously been aware of the debt (i.e. he did not get notice of the Court hearing where the liability order was granted, or the other letters that will have been sent) he ought to be able to get the Council to withdraw the case from the bailiffs if he makes an acceptable payment offer.