Keep in mind that the camps are separated fairly quickly - the Shipburning marks a decision on the part of the Fëanorean host to go to Middle Earth without the (larger) host of Fingolfin. They have no reason to suspect that the host of Fingolfin will ever arrive in Middle Earth; they have to assume that they will go back to Valinor.
So, at that point, the entire host of Fëanor dismisses the significance of Fingolfin, and it is 'obvious' that one of Fëanor's sons would lead in the event of his death. I think we will have reason to talk about that, as Fëanor seems quite mad/fey from the point of the Shipburning on (I mean, even more so than he does during the Oath/Kinslaying/Doom of Mandos). Also, Maedhros breaks with his father over the decision of the shipburning and does not take part, showing a rift between the two. It's not a complete split, but it's the type of incident that should have a ripple effect, with people wondering if Maedhros is going to rebel against his father's leadership at some point. If the Fëanoreans have factions, Maedhros surely leads one of them.
And so when Fëanor does die in battle, the only people around to assume leadership are his sons. And...that's not as clear-cut as you'd think. Maedhros does not claim the title of High King, and when he *does* decide to meet with Morgoth's emissary, he does so *without* his brothers. So, was that a decision he made that they disagreed with, or...? Who is really in charge and calling the shots here? Surely the soldiers Maedhros took with him to that meeting were his faction, his most loyal followers - and Morgoth's balrogs slaughtered them all. Maedhros is the only elf who survives that meeting.
And so the other Sons of Fëanor are left behind, being told that their brother is alive but with no proof of that. It is somewhat assumed that Maglor is in charge, but that's not really clear, and the Fëanorean camp seems to be rather confused and lacking in leadership at this time. They don't do much of anything until....
Fingolfin's host arrives, having marched across the Ice. They are pissed off at Fëanor, who betrayed them and left them behind on the shores of Valinor, and they are a much bigger army than the (remaining) host of Fëanor. No one in Fingolfin's camp would want to submit to the leadership of any of the Sons of Fëanor (and people like Turgon are going to be very vocal about this).
So, you have the host of Fingolfin thinking that Fingolfin is the natural leader of all the Noldor, and the host of Fëanor thinking that one of the Sons of Fëanor should be in charge, but maybe having some doubts now that Fingolfin has arrived. So, like any reasonable people, they address this issue by studiously not talking to each other.
Enter Fingon - he realizes that Maedhros is missing, so he goes to find him. Presumably, he just misses him, but there has to be some residual thought of 'together to two of us can fix this mess that our fathers got us into.' And they do. After Maedhros returns, obviously owing his life and freedom to Fingolfin's son, he has a perfect opportunity/excuse to acknowledge Fingolfin's claim to the crown himself. And as Fëanor's eldest son, he immediately quashes the claims of any of his brothers. (His declaration emphasizes that Fingolfin is the eldest of the House of Finwë who is still present - which of course emphasizes his own right to speak for his brothers. Maedhros is the statesman of the House of Fëanor!)
...at least, that is one way of looking at this. Political storylines are not my strong suit, so.