If by the early 80's any member of the Provo's had seriously put forward a plan to assassinate the Queen the first thing that would happen would be they would be told, quite firmly "No" by everyone above them.
The second thing is that MI5 would have been informed about the plan as quickly as humanly possible because between the informers and individuals with connections every action would be taken to keep her alive.
The Provo's, while I wouldn't call them "smart", were canny bastards and knew that killing the Queen would be crossing a line they could never come back from. Even killing the PM wouldn't be as bad as that. The only time I could see them seriously considering it would have been at the absolute height of the anger in the Troubles, Bloody Sunday, and even then for it to actually have happened would require so many things to go perfectly "right" that the odds are astronomical.
So, all that out of the way, I see it working out somewhat like this:
Queen dies, probably takes Philip with due to the IRA not exactly being precision experts in an operation that would be needed for this. The moment it is confirmed to be the IRA (or more likely a splinter group there of) it becomes open day on anyone Irish in the UK, depending on the date, that means odds are my Father winds up in prison for being Irish and my Mother gets subjected to all the abuse in the world, up to possible violent and or sexual. To say it would be an outpouring of grief and anger would be charitable, the reality is that a large portion of society would be out for blood.
After that outbreak of violence, if the perpetrators have not already been killed, the second the IRA can get their hands on them they offer them up tied in a fucking bow to the British authorities, but that will not be enough. Thatcher will be already facing loud calls to resign for failing to keep the Queen safe, so to try and keep her office she'll double down and order the sort of crackdown in Northern Ireland that only the most delusional of Loyalist would want. It won't work, but it will satisfy the need for blood.
Speaking of the Loyalist community, the word "pogrom" comes to mind to describe just the level of state sanctioned violence they would now be forcing on the Nationalist community, expect families to flee to the Republic with little more than the clothes on their back and for the Bogside to resemble Cork in 1920.
The bombers won't be able to be executed, the law being as it is, but they will be subjected to the absolute most inhumane treatment that the British system can come up with, and used as a prime example of why the Irish can not be negotiated with whenever anyone on the international stage broaches that maybe Britain should calm down a bit.
In the Republic, the shit hits the fan. Almost any request for an extradition in relation to perceived IRA crimes would be quietly acquiesced to, simply due to the fact the alternative would not bare thinking about. A massive refugee wave from the North coupled with the already desperate state of the economy in the 80s will see Ireland become the basket case of the EEC and probably not able to meet the obligations to join the Eurozone when that gets going. Even Greek level book-cooking wouldn't manage that. Probably we wind up being the poorest country in the EU until such a time as Bulgaria joins, and even then...