A reverse image search reveals this website as the source of those two images. It belongs to the 7th United States Infantry living history association.

The "wedge-style" hats, as you call them, are more commonly known as forage caps. This particular type was common in continental European armies from at least the late 18th century onwards.

The peaked or wedged forage cap is originally a long peaked woolen cap with a bobble, pom-pom or similar at the top (think Santa Claus or nightcap). The peak has been folded into the brim of the cap so that only the pompom protrudes at the front. This type of military cap has since evolved further, eliminating the folding long top, while retaining the original folded wedge shape. This modern type of military headgear is known by various terms, including garrison cap, side cap, boat cap, wedge cap, calot and more.

During the period in question, this type of cap would be used as an off duty or downtime cap, as it was usually much lighter and more comfortable to wear than the full dress regulation shako, helmet or similar headgear. It would be worn in camp or garrison when performing light duty or not otherwise arrayed for battle, parade or full drill.

When it comes to the units depicted, these are obviously reenactors, which makes it difficult to speak to the accuracy of the depiction without diving into period regimental histories, rosters and regulations. As another poster has commented, the origin of the photos appear to be an American living history association primarily concerned with recreating unit(s) from around the time of the Battle of New Orleans and the War of 1812, which means the uniforms are American in origin. If so, then the blue coated soldiers would be infantrymen and the red coated soldiers regimental musicians of the 7th US Infantry Regiment as they appeared in 1814/15 (provided their recreation is accurate).