Escalating means getting more intense. Many organizations use this to get more intense on you! They say “if you don’t fix it right now I have the power to escalate!” This is threatening, isn’t it.
Well let’s get it simple: there’s no threatening affecting an external consultant.
Client can escalate internally, but this is not you problem. Be calm and keep prompt but firm. Their boss is not your boss. Escalation never made fixes happen. It rarely worked with lazy people to speed up their activities.
So, if you are not the problem cause and you are not lazy then there’s no effect on esclating. Keep focused on defining what the blocking issue is and where. Leave others wasting time on “escalating”.