Do nose cones work well at high angles of attack?
No, they don't.
The slanted intake is slanted because the top part produces a compression shock wave that hits the lower intake lip so all of the compressed air is ingested and nothing is wasted. In order to do this reliably up to Mach 2.5 (where the Mach cone angle is 73.8° from the vertical), the whole intake needs to be slanted by those 73.8° (actually a bit less due to the intake wedge). In order to adjust the intake for different Mach numbers, one or both parts are adjustable, as shown in the F-14 example below.

Cross sections of the F-14 intake at subsonic, transsonic and supersonic speed. Note that the shocks of the supersonic example do not fit well with the intake geometry – the geometry is optimized for a much higher Mach number than shown. See here for a better example.
Here is (admitted a quite messy) cross section of the F-15 intake. The adjustable upper wedge is clearly visible (picture source):

In order to minimize losses, the upper intake uses adjustable ramps in order to force additional shocks so the compression happens over a sequence of weak, oblique shocks.
Regarding spike intakes: They were the lightest and simplest way to produce efficient supersonic intakes and were ill suited to large sideslip angles or angles of attack. Note that they were a characteristic of early supersonic flight when speed was the main objective and have since been replaced by ramp or even pitot intakes.