It’s something i have been thinking about. All the other major nations in Europe had their own AT launcher of some kind. Americans with the bazooka, British/commonwealth with the PIAT and the Germans with the panzershreck or panzerfaust. I know that they had the PTRS AT rifle as well as AT grenades but wouldn’t it make sense to try and make their own rocket launcher to fight such a Mechanised foe? Did they just feel that their tanks, and AT grenades were enough? I do realise that they used allied weapons but what led to the decision to not invent their own until much later?
The USSR developed a number of anti-tank rocket launchers during the Great Patriotic War, but none of them were put into production. The problem wasn't with the launchers themselves, but rather the army's unrealistic expectations of them.
You mentioned the American Bazooka, a fine weapon that did very well in WWII and afterwards. The USSR received a number of these weapons as a part of Lend Lease aid. Trials were performed, but the Red Army didn't like them. Like all rockets, they worked poorly in cold temperatures, the muzzle velocity didn't allow them to hit a moving target at any appreciable range or even a stationary target at over 100 m, and the backblast was quite a big problem when firing from a trench or dugout. The Red Army wanted a weapon that could completely replace the PTRD and PTRS, meaning that it could hit a tank sized target at several hundred meters and could be fired from any location. Only 3000 Bazookas were received, they never saw combat, and in 1943 the GAU asked the Soviet government to not order any more.
Domestically developed anti-tank rocket launchers popped up as early as 1941, but they all butted up against this requirement. Generally they revolved around firing RS-82 air-to-ground rockets from a land-based launcher. The precision was low and they couldn't hit a target at 200 meters (to be fair, no wartime anti-tank rocket launchers could do this). Precision could be improved by rotating the rocket, but that resulted in a drastic reduction in penetration of the HEAT warhead. The ground based launchers also had shorter rails than wing-mounted ones, which meant that the trajectory would be different and they won't fly as true.
That being said, it's not like Soviet troops didn't use rocket launchers. Captured Panzerfausts were available in great quantities, and sappers used various rockets held in wooden frames to fire at point blank range against German fortifications.
A satisfactory weapon was finally tested in 1944: the RBG-82. This weapon had satisfactory precision at a range of 300 meters, but only in the summer, in the winter the classic issues exhibited by all anti-tank rocket launchers began to show. It took until the late 1940s to iron out all of these issues, by which point the war was long over.
To summarize, the USSR produced a number of promising anti-tank rocket launcher designs that were just as good if not better than their foreign analogues, but Soviet requirements for these weapons were also much higher.
http://www.tankarchives.ca/2017/08/lend-lease-impressions-bazooka.html
http://www.tankarchives.ca/2019/10/lend-lease-wishlist.html
http://www.tankarchives.ca/2014/12/rocket-gun.html
http://www.tankarchives.ca/2016/05/world-of-tanks-history-section-new-life.html
http://www.tankarchives.ca/2014/10/panzerfaust-russian-style.html
http://www.tankarchives.ca/2014/12/rocket-bike.html