As there's no answer for a long time, I've done some research with help of the links provided by others in the comments to the questions.
And the "first" goes to ... Soviet Union.
In short: US got a head start with the blunt object theory, but in terms of heat shielding got diverted to less efficient "heat sink" path, wasting some time before employing ablative shield; whilst USSR worked with sharp cones and, having operable boosters made available earlier than US, worked through the hard way of experimentally burning the payload in atmosphere and choosing ablative shielding straight away.
So, here's the timeline:
July-September 1955 : USSR experimental version of R-5 rocket source in Russian:
Экспериментальная ракета М5РД была разработана на базе конструкции Р-5 для проверки в лётных условиях ряда новых систем и принципов, разработанных для межконтинентальной ракеты Р-7
Все 10 пусков, проведённые в июле-сентябре 1955 года с аппаратурой системы управления Р-7 были успешны.
Во время трёх из 10 осуществлённых пусков М5РД испытывались головные части: одна с покрытием из карбида кремния и две с асботекстолитовой теплозащитой.
The M5RD experimental rocket was developed on the basis of the R-5 design to verify in flight a number of new systems and principles developed for the R-7 intercontinental rocket. All 10 launches carried out in July-September 1955 with the equipment of the R-7 control system were successful. During three of the 10 completed M5RD launches, reentry vehicles [literally translated from Russian as "head part"] were tested: one with a silicon carbide coating and two with asbestos-textolite thermal protection.
Ablative nature of used asbestos-textolite is verified here
8 August, 1957: USA; Jupiter-C, subscale reentry vehicle with ablative nose-cone.
Source: Coming Home. Reentry and recovery from space (I highly recommend this very interesting book, special thanks to @WayneConrad for the link)
While the U.S. Air Force and its contractors were expending efforts on heat sinks, the U.S. Army was taking the lead in investigating ablators.
On August 8, 1957, a Jupiter-C launched from Cape Canaveral carrying a subscale reentry vehicle to an altitude of 600 miles and a range of 1,200 miles and proved the feasibility of the ablative-type nose cone during reentry.
It should be noted, however, that this test was only meant to validate the reentry conditions for an intermediate-range Jupiter warhead. Whether an ablative ICBM warhead would work was still unproven, although in theory the concept could be scaled up for those more advanced systems. The Air Force initially looked upon the Army’s claimed success with skepticism...
The NACA researchers proved the Army was right: ablative coatings were superior to heat sinks.
21.08.1957 (М1-8/8Л) and 07.09.1957 (М1-9/9Л): USSR First and second successful launches of R-7 ICBM rocket. Both Reentry Vehicles (RV) burnt up in the atmosphere on reentry.
I.S. Prudnikov, one of the leaders of the team that was working on R-7 heat shielding, confirms that both flights were using ablative asbestos-textolite heat shield. He also describes post-flight analysis:
В результате анализа полученных данных удалось уточнить величины уноса теплозащитного покрытия, а по ним - методики расчета. Это обусловило изменение аэродинамической компоновки - была уменьшена длина наконечника, увеличены его притупление и длина стабилизирующей юбки
As a result of the analysis of the data obtained, it was possible to update the values of the heat-shielding coating ablation, and from them the new [theoretical] calculation methods. This led to a change in the aerodynamic layout - the tip length was reduced, its degree of blunting and the length of the stabilizing skirt were increased
Left- original design (1956); Right- updated post-flight design (1958)
In the meantime, third and fourth successful R-7 flights were used to launch Sputnik and Sputnik-2 whilst the mishap with RV's burning on reentry was being fixed.
30.01.1958 (М1-11): USSR, fifth successful R-7 flight (with updated ablative heat shield). RV failed to separate from the rocket's second stage, but both made it to target area.
29.03.1958 (М1-10): USSR, sixth successful R-7 flight (with updated ablative heat shield). Fist time the RV reached the target without burning up in the atmosphere.
9 September 1959: USA; Big Joe 1 (Atlas-D), first test of Mercury capsule (boilerplate) ablative heat shield.
Atlas 10D was built originally as an R and D vehicle but had received the initial Mercury modifications. The payload was a boilerplate spacecraft. The purposes of the flight were to test the spacecraft's ablative heat shield, afterbody heating, reentry dynamics attitude control and recovery capability.
And from here
Although the range was less than planned due to failure of the Atlas booster engine section to separate, the capsule was recovered in the South Atlantic after surviving reentry heat of more than 10,000°F.
6 October 1959: USA, earliest launch of ablative Mk3 RV atop Atlas-D that I could find (Mk1 and Mk2 were heat sink). Mk 3 Mod 1 re-entry vehicle research and development mission.