Some nations did develop and issue armour during both wars, and there was definitely armour improvised by troops in the field. I can provide two examples, both from WWII. There are other examples from both WW1 and WW2, I'm just not familiar enough with them to provide a detailed answer.

1)American Aircrew: some American bomber crews were issued body armour to protect them from splinters from exploding Anti-acircraft shells and cannon rounds fired by enemy fighters. The armour was fairly heavy and bulky (see photos at link), but that was less of a concern for aircrew manning fixed positions inside large multi-engine bombers. (source: USAF Museum https://www.nationalmuseum.af.mil/Visit/Museum-Exhibits/Fact-Sheets/Display/Article/1519644/bomber-crew-protection/)

2. British Personal Armour: The British Medical Research Council developed personal body armour starting in 1940. The final version consisted of 3 thin steel plates suspended in a webbing carrier. The plates were situated to protect the vital organs. It was issued to some army units, including the infantry units and artillery forward observation teams of 2nd Canadian Infantry Division before they were deployed to Normandy shortly after D-day in 1944. In his excellent book, The Guns of Normandy, George Blackburn relates the story of at least one Canadian Officer whose life was saved when it stopped a burst from a 9MM Submachine Gun. Blackburn also notes, however, that many men chose to discard the armour, as it was hot, uncomfortable, and there were doubts about it's effectiveness (this is taken from Chapter 26, The Guns of Normandy, by George Blackburn). Milart has a well written and thoroughly researched article on the subject that goes into much more detail, and also features some excellent photos of the armour in use by field troops: https://milart.blog/2014/07/02/british-medical-research-council-mrc-body-armour-in-canadian-use/

I will refer to an answer I've previously written on this topic, which is short enough that I'm going to quote it in full here:

There were experiments with body armor during World War II (and even some during WW1), most famously including the Russian SN-42 steel engineer's breastplate. The SN-42 and similar steel cuirass designs were, broadly, protective against fragmentation and submachine gun fire at combat distances. Soviet soldiers reportedly found them beneficial in urban fighting, but heavy and uncomfortable. I am also aware of American experiments with aluminum-based plates, including aluminum/Doron fabric laminates, mostly as an avenue to improve aircrew protection over the ballistic nylon flak jackets. It would seem some of these designs could resist fragmentation & splinter threats similar to .30 carbine. However, I am not aware of these plates being fabricated into vests for air or ground crew issue.

None of these armor types were resistant to full-caliber rifle fire at combat distances. While it would certainly been possible, from a metallurgical perspective, to produce steel armor capable of stopping .30-06, 7.62x54R, etc, it would have been prohibitively heavy. American studies for the Variable Body Armor program during Vietnam concluded that the maximum useful density for infantry body armor was about 7-7.5 pounds per square foot; beyond this, it's just too heavy and fatigues the poor bloody infantry too much. For comparison, modern AR-500 armor steel is about 12 lb/sq.ft. There are modern armor steels that surpass AR-500, of course, but as far as I am aware those were mostly developed during vehicle armor research during the Cold War.

There was also research before and during the war on ballistic nylon armor, as used in flak jackets, but again - this stuff is good for fragmentation and possibly pistol ammunition. As far as I am aware, nobody issued it to ground troops, likely because early ballistic nylons were heavy, stiff, and hot - traits which are okay for bomber crews, but not so good for Private Snuffy the rifleman.

Sources:

Crouch, Ian G. The Science of Armour Materials. Duxford: Woodhead Publishing, 2017.

"Resume of Programs on Development of Body Armor Undertaken at Watertown Arsenal During World War II." Defense Technical Information Center, Accession Number : ADA954943. https://apps.dtic.mil/dtic/tr/fulltext/u2/a954943.pdf