They have 28 days from the date that the ballot result was announced in which to commence strike action, but need to provide BA with 7 days notice of any proposed strike dates. Assuming that the strike action is protected (i.e. it is not seen as a continuation of an existing dispute) the union has a period of 12 weeks in which it can call for strike dates. Thereafter, protect ceases to be in place and any industrial action taking place is not protected. The protection gives employees the right to claim for unfair dismissal if they are sacked solely for taking part in industrial action. It also provides some indemnity for the union.
The problem that Unite face is that the 10 items in their list of grievances include wording that inextricably links this lates ballot with the previous strikes, hence it is likely that any industrial action would be unprotected. Whilst BASSA and co may wish to call strike dates, they are powerless as only the union is able to do that. If Unite feel that in doing so they would open themselves up to substantial legal action, not just from BA but potentially from the membership for not providing the whole truth about the legality of the current situation, they may well back down.