The manufacture of assault rifles needs several technological advances over mid-nineteenth century weapons:
1) Readily available machined-steel parts (barrels, bolts, springs, etc.). Wrought iron won't do; it is too susceptible to metal fatigue. The relevant tech was not widely available until the 1890's OTL.
2) Smokeless powders. Black powder fouling would necessitate cleaning the barrel every 20-30 shots to prevent jamming and worse problems, which would make the rate of fire no better than a magazine-fed breechloader, and the latter weapon is more accurate. Such powders were not widely available until the 1890's OTL.
3) Brass cartridges. While it is possible to use paper or cloth cartridges, they are susceptible to moisture and more vulnerable to damage. Brass cartridge weapons were just coming into use in the 1860's and were not widespread until the 1880's.
That just makes the weapons technically possible; to get them built there has to be a perceived need for them as well. IOTL that did not happen until WW II, and true assault rifles did not appear until 1944 (StG44).
Why not earlier? Let's review the history. Prior to the 1850's armies used single shot muzzle-loading smoothbore weapons. They did so because breechloading weapons were inefficient and unreliable; the close tolerances necessary for proper breechloading weapons were simply not possible with the methods and materials of the time. The weapons were smoothbore because they could be loaded faster and were less susceptible to fouling than rifles. With these weapons the effective range was no more than 150 yards and volume of fire counted for more than accuracy, which meant large masses of troops firing at close range into other masses of troops.
The Minie rifle and its successors changed that. Fully as fast to load as a smoothbore, accurate to 300 yards or more, and less susceptible to fouling (because the tighter fitting bullet tended to scrape residue out of the barrel when it was fired) they were immediate successes. Troops using linear tactics were slaughtered facing these weapons (casualty rates in ACW battles regularly exceeded 33% of the troops involved). By the end of the Franco-Prussian War most people had gotten the message; frontal attacks were eschewed and troops took full advantage of available cover.
The advent of breechloading rifles merely reinforced this lesson; they were accurate to even longer ranges (as much as 1000 yards), could be loaded from a prone position, and had much higher rates of fire than muzzleloaders. Magazine rifles offered increased rates of fire, but the basic principles remained the same as for Minie rifles.
Until the later stages of WW I the tactical thinking remained the same as at the end of the Franco-Prussian War; massed aimed rifle fire was king. Defense was superior to attack; strategy consisted of maneuvering to position one's force where the enemy would have to attack it in circumstances favorable to the defender.
In those circumstances what advantages would an assault rifle offer to the user? Since engagements were expected to be at long range, the shorter effective range would be a handicap. Ammunition supply would be a problem, especially in the opening stages of the war when manufacturing had not yet increased to meet the needs of existing weapons, let alone these. The different type of ammunition used would be a further handicap (most machine guns of the period used the same ammunition as the rifles). Given the tactics of the time such weapons would actually be detrimental to the users.
IOTL assault rifles were developed to fill a perceived need. In the case of the Germans, after the aerial assault on Crete demonstrated the inadequacy of existing paratroop weapons; in the case of the Russians, after observing the superiority of German small arms to current Russian weapons.
Assault rifles will not be developed until the technology can support them and the perceived need for them exists. Neither is really possible before the late nineteenth century without a very early POD (or PODs).