In the present circumstances is there any order of priority in which the Queen would call potential Prime Ministers to the Palace?
This is answered in the Cabinet Manual mentioned in user's answer. Some relevant quotes:
In modern times the convention has been that the Sovereign should not be drawn into party politics, and if there is doubt it is the responsibility of those involved in the political process, and in particular the parties represented in Parliament, to seek to determine and communicate clearly to the Sovereign who is best placed to be able to command the confidence of the House of Commons. As the Crown’s principal adviser this responsibility falls especially on the incumbent Prime Minister, who at the time of his or her resignation may also be asked by the Sovereign for a recommendation on who can best command the confidence of the House of Commons in his or her place.
Under the Fixed-term Parliaments Act 2011, if a government is defeated on a motion that ‘this House has no confidence in Her Majesty’s Government’, there is then a 14-day period during which an alternative government can be formed from the House of Commons as presently constituted, or the incumbent government can seek to regain the confidence of the House. If no government can secure the confidence of the House of Commons during that period, through the approval of a motion that ‘this House has confidence in Her Majesty’s Government’, a general election will take place.
Where a range of different administrations could be formed, discussions may take
place between political parties on who should form the next government.
In other words, it is the responsibility of the outgoing Prime Minister to recommend to the Queen who she should invite to form a new government. The PM's recommendation may be delayed until any discussions within and between parties have made some progress.
This happened (albeit under different circumstances) in 2010, when Gordon Brown stayed on as PM for a few days after losing the election, until it became clear that the Conservatives and Lib Dems were about to form a coalition with a majority in the Commons.