Let's be honest, the EU needs the UK's decision to have some clear negative consequences. I think Merkel is being clever. Let's think what will happen if the UK delays invoking article 50. There will be an extended period of uncertainty, during which time investment will drop and the economy will likely enter recession. People will not like this, especially the leave supporters who will suffer most. Farage will tell us that it's a price worth paying and that good times are ahead, but as people lose their jobs and see living standards fall, they may start to disagree. The UK will suffer and views towards EU membership in the UK will likely become more favourable.
This is not the say that the UK will reverse its decision, but other countries will be watching carefully, and the UK's self inflicted suffering will be the best advert possible for continued EU membership.
The UK will delay invoking article 50. It will delay at least until the Tory party has a new leader, and there is a clear plan for what comes next. After that, the UK government will try to start unofficial negotiations with other EU leaders, but they may well refuse, telling the government to listen to their people and invoke article 50. But having invoked article 50, the UK has little negotiating power - the rest of the EU merely has to wait for the two year notice period then they can impose more or less what they like.
All this will happen during a constitutional crisis in the UK with serious problems in Scotland and Northern Ireland, and economic problems.
I wonder if the government will ever invoke article 50.