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In keeping up to date on Brexit, Dominic Cummings, Special Advisor to Boris Johnson - UK Prime Minister - has stated that, in the event that a motion of no confidence has passed the House of Commons, Johnson could ignore a motion of no confidence, even if an alliance, holding the majority in the Commons, was formed:

Cummings is said to have told advisers that Johnson would ignore the result of the confidence vote and call a “people v politicians” general election – to be held after Britain had left the EU.

Reference

To my mind, the Fixed-term Parliaments Act of 2011 doesn't make provisions for a now defunct Ministry to pick a time a General Election is to be held without the express consent of the Commons. However, as stated be Cummings, the current Ministry intends to attempt to do just that while not recognizing the formation of a new Government, if a new Government had been formed:

Cummings, the former campaign director of Vote Leave, is understood to have told government advisers last week that Johnson could stay on as prime minister even if rebel Tory MPs were able to form a “government of national unity” opposed to a no-deal Brexit.

My interest is not in the legality of whether they could do such a thing, but, in the event Her Majesty is brought in to resolve the issue, what does Her Majesty's Prerogative allow in the such an event where the previous Ministry (the Ministry that lost the no confidence vote) ignores both the no confidence vote and the guidelines pursuant to the Fixed-terms Parliament Act while a formal alliance holding a majority in the Commons is presented to Her Majesty?

And second to that, does Her Majesty hold the power to order the Police, Army, or even the Grenadier guards to arrest the individuals making up the defunct Ministry in order to allow the new alliance to form a Government?

Her Majesty is obliged to appoint as Prime Minister the person who she feels is most likely to command the confidence of the House of Commons.

If Boris Johnson loses a vote of no confidence (as set out under the Fixed Term Parliament Act), and another leader is able to convince Her Majesty that he is more likely to be able to get the confidence of the Commons within that two week period, then Her Majesty would be within her rights to sack Mr. Johnson and appoint this other person as Prime Minister. While the sacking of a Prime Minister has not been done in the UK (at least not in modern history), it was done in Australia in 1975, in what became known as The Dismissal.

After this two-week period, the election must be held, but as the Fixed Term Parliament Act does not specify precisely when said election must be held, it is up to Her Majesty on the advice of the Prime Minister. It would be expected that there would be a short wash-up period to get some essential business through (though said business would require support from the opposition), and then Parliament would get dissolved five weeks before the election.

Her Majesty can appoint whomever as Prime Minister. By convention, that is on the recommendation of the outgoing PM, but the recommendation is not required.

Having been appointed, the Prime Minister can organise their Cabinet however they desire, re-arrange the Civil Service, etc.

Johnson barricading himself in Downing Street would be a futile gesture, (presumably) Corbyn could govern from some other building.