the hijacked portion of overrunning 1B and missing the base...

[Tru_in_Blu]

BR is running to 1B and the throw from an infielder is low and wide to the HP side of 1B. BR sees the bad throw and attempts to avoid being hit, or is actually hit by the ball, causing her to miss 1B. As she runs past, she loses her balance and falls onto the grass down the line. There is no obstruction by F3 and no interference by BR.

The BU dutifully declares the runner "safe!" as we are supposed to, incurring the rath of many uninformed fans, and hot moms.

Everyone on the field knows the base was missed. BR scrambles to her feet and darts back to the base. F3 retrieves the ball that went to the fence while F4 covers 1B. F4, standing on the colored portion of the base takes the throw from F3 a split second before BR slides back to the white portion of the base.

As BU, do you make a call right then? Or do you have to wait for someone to "appeal" the missed base? If the out would have been recorded based on the play, can someone appeal after the fact and before next pitch? Since everyone knows what's happening, does the fielder with the ball need to declare an appeal for a missed base while actively trying to deal with covering the base, catching a throw, and a sliding runner?

On a "left too early" call on a fly ball when a runner is trying to get back to a base and doesn't get there in time, umpire will just make the call without a verbal declaration from the defense. Similar or different for overrunning/missing 1B?