In his first appearances, Captain America's shield is an elongated triangle with three peaks at the top:
Joe Simon's first sketch of Captain America
Cover of Captain America issue #1, 1941
Now, of course, his shield is round.
Late 60's issue of Captain America
When did this change occur, and why?
I cannot find any in-universe reason as to why this occurred, but out-of-universe I can.
Apparently the original sketch of Captain America came to the attention of Archie Comics (then MLJ Magazines) - they noticed there was a similarity between Captain America's shield and the chest detail of their own character The Shield (below).
Consequently, Captain America's shield was changed to the Kite design.
(Source: Cronin, Brian (2009). Was Superman a Spy?: And Other Comic Book Legends Revealed. Plume. pp. 135–136. ISBN 978-0-452-29532-2.)
The Marvel Wikia site refers to President F D Roosevelt presenting Captain America with this. This puts it between 1933 and 1945
From Captain America Vol 1 255: 'The Living Legend' in which
It details the origin of Captain America and focuses on his early adventures in 1941-42.
As correctly pointed out in the question:
Rogers was originally issued a traditionally "kite" shaped shield made of mundane steel, as well as a sidearm.
(Source)
This new disc-shaped shield, however, had two main advantages:
1: It was better as an offensive weapon
He discovered:
that its excellent aerodynamic properties made it an effective offensive weapon
(Source)
2: It was made of a nearly indestructible alloy
Originally, Dr. Myron MacLain was attempting to perfect an indestructible alloy as strong as the legendary 'Adamantine', which Hercules' Golden Mace was made from. Dr. MacLain was hoping the alloy would provide a distinct advantage in the armor of American war machines, during World War II. The Doctor worked tirelessly, using the exotic Vibranium metal and an unknown substance. While the Doctor was asleep, as a result of his exhaustion, an unknown factor caused the metals he was working with to bond.
(Source)
An almost indestructible shield has obvious advantages over one made from 'mundane steel'!
Not only is the shield nearly indestructible though, it also has other useful properties which are inherent to Wakandan Vibranium, namely the ability to absorb vibratory energies in the vicinity.
The Wiki page on his shield explains the following (however bear in mind that it does not reference the said sources):
Some sources say that Dr. MacLain chose the discus shape because of its versatility, while other say that he used an existing cast designed for tank hatches.
In-universe, Captain America got round shield in 1941. See the middle-right panel of this page from Avengers (1963) #71:
The same panel also shows why Cap wanted to get a round shield.
Note: This comics doesn't give any out-of-universe cue. When the comics came out, Cap already had round shield. In this scene, three Avengers (Black Panther, Yellow jacket and Vision) are in past when Cap didn't know anything about the Avengers.
Update:
Out-of-universe, Captain America first appeared with round shield in Captain America comics #2 (Published: April 01, 1941). You can see it on the cover:
The first page:
In the previous issue which was published on March 01, 1941 (whose cover you've included in the question), Captain America kept using that kite/badge shaped shield until the end.
They suddenly changed the design of the shield without giving any in-universe explanation at the time.
Captain America's first shield was badge-shaped. After complaints by a rival comic company that it too closely resembled their hero's symbol, Captain's shield was replaced in Captain America #2 with a disc shaped version.
