I'm sure someone can give a more detailed (and better formatted) answer than this, but the initial reaction was of some concern, albeit quite delayed for understandable reasons.

The USSR adopted the AK-47 in the year 1949, but it was not widely issued or exported/donated to Soviet/Warsaw Pact allies until the early-mid 1950s due to production delays. Most Soviet Army units were issued the SKS rifle (adopted in 1945, and in some sources deployed in a limited fashion at the very end of WW2), although the so-called front-line units possessed it only briefly due to the adoption of the AK-47. The most obvious impact of the AK-47 (and later AKM) can be seen in the development of the AR-15, which would in turn be slightly redesigned and adopted as the M16, XM16E1 and then M16A1 rifle between 1964 and 1966, but the initial time-gap here is telling for a few reasons - namely, the AK-47 was not widely deployed by the USSR or Warsaw Pact in combat operations. Egypt had some during the Suez Crisis (1956), and the small number of examples captured by Anglo-French and Israeli forces were quickly handed over to the SIS, SDECE and then the CIA, but given the recent adoption of the FN FAL (1958, ubiquitous in Europe and the wider Western Bloc by 1961) and M14 (1959) rifles and the fairly limited fighting during Suez, academic concerns were raised but no major concerns were addressed - the FN FAL and M14, and slightly later G3 (1964, replacing the FN FAL in Bundeswehr service and exported with some success later) were all selective fire too, and performed adequately in both urban and desert warfare - the latter's high power, heavy cartridge being a noted advantage across the relatively long distances inherent to the latter theatre.

Much of this began to change in the mid-1960s, as US involvement in South-East Asia (primarily Vietnam) began to escalate after 1965. At the sort of close range fighting inherent to the jungles of Indochina, the M14 and older M1 Rifles issued to the ARVN could be outgunned by the AK-47, and older Soviet weapons such as the high-capacity PPSh-41/PPS-43. This is simply because that while the older US weapons served adequately and could boast high-powered, hard hitting cartridges in their 7.62x51mm and 7.62x63mm calibres, and therefore also had very good cover penetration, the AK-47 also enjoyed a lighter (albeit still somewhat heavy) round in it's 7.62x39mm calibre. Most crucially, the M1 and M14 rifles were issued with 8-round clips and 20-round magazines respectively. The M14 was also typically fed with stripper clips rather than a multitude of detachable magazines, and there were some concerns regarding the power, length and weight of both rifles when issued to the generally smaller ARVN troops. The M1 and M2 carbines (7.62x33mm) were a partial solution to this issue, but it was considered underpowered. The AK-47 fired from 30-round, robust magazines and enjoyed a more-controllable recoil than it's adversaries. It was highly inaccurate at medium ranges in combat conditions, but this was of lesser concern in jungle and forest combat, and it's reliability was legendary. ARVN troops and their advisors could, and did, occasionally report that they were losing firefights under the conditions they fought in. US formations were more immune to this danger, but not fully.

Regardless, by 1965 concerns had become known to the US Military's highest ranks, and requests for new weapons had begun to circulate. The US Airforce had in 1966 issued a request for a new rifle in .22 calibre, and Armalite had by then reworked Eugene Stoner's earlier AR-10 (7.62x51mm) into the AR-15 (5.56x45mm). The weapon's early history as the M16 was deeply troubled due to a number of flaws regarding it's reliability in battle, but the calibre was not one of them despite early misgivings, and these were mostly solved by 1968 and the issue of the new M16A1 rifle. US troops now, generally speaking, had a weapon which could outgun the AK-47 at almost all ranges by virtue of it's lighter ammunition, which in turn allowed more to be carried. The early 20-round magazines were largely replaced by 30-rounders in 1969, though the M14 was retained by some second-line Navy and Army supporting units, primarily Engineers, well into the 1980s. The AK-47 also proved of great worth against the FN FAL in the Six Day War (1967) fought by Israel against a coalition of Arab states despite the rapid Israeli victory - that legendary reliability of the AK showed itself in stark contrast to the Israeli version of the FN FAL, which carried a faulty dust-cover and easily allowed sand to jam the weapon if poorly maintained. The M16 quickly became standard for a number of NATO militaries and other US Allies, though the FAL and G3 were retained by some for decades and won good combat records despite some drawbacks in various theatres. 5.56mm became the NATO-standard calibre and was ubiquitious by the 1980s, though the initial US version of the round did not due to it's excessive powder-fouling. The USSR introduced the AK-74 in it's new 5.45X39mm calibre in the late 1970s as a response to reports of the AK-47/AKM's performance in Vietnam and Indochina more widely, though many second-line units retained it well into the 1990s.

TLDR - The AK-47 was an excellent weapon for it's time, but all weapons systems age and by the late 1960s it's flaws were well known, even if the weapon itself wasn't exactly obsolete.