Both of these crimes were admitted by camp commander Max Pauly in his deposition of July 3rd, 1946 (NO-1201, p.5), which was made after Pauly had already been sentenced to death. Pauly claimed that gassing and injections were ordered by Amtsgruppe D whilst the hanging of the children was ordered by Oswald Pohl. Pauly also stated that in 1941-42, Lolling ordered the transfer of insane patients to Bernburg euthanasia centre "for their execution" (p.6).
The gassing of the POWs was carried out by Willi Bahr, who gave this testimony on the stand at his trial in response to questions from his own defence lawyer. He had given an affadavit about the gassings on November 30, 1945 (see here), cited in the Tesch trial (case summary, pp. 95-96), and the gassings had earlier been the subject of testimonies by Eduard Zuleger und Günther Wackernagel on August 8th, 1945 (see here). Bahr was executed on October 8, 1946. As Steve Tyas notes here, Bahr's guilt is also tied to documentary proof of supplies of Zylon B to the camp:
In the year 1942, KL Neuengamme purchased 180 kgs of Zyklon B (3 March-18 Dec 1942) for 1,108 RM.
In the year 1943, KL Neuengamme purchased 427 kgs of Zyklon B (21 Jan-31 Dec 1943) for 2,508 RM.
All these purchases were from Tesch und Stabenow, Hamburg, compiled from original T&S documents used at the trial of Tesch, Weinbacher and Drosihn. The first two were convicted and hanged at Hameln; Drosihn was acquitted.
The experiments on the children were conducted by Kurt Heissmeyer. Owald Pohl gave this handwitten account (NO-065) of how the children were transferred from Auschwitz:
5. Heissmeyer - An Oberarzt (head physician) in the medical institution in Hohenlychen, received from Himmler permission to carry out experiments with tuberculosis. I sent him to Gluecks who put the necessary persons to experiment upon at his disposal. He received about ten orphan children who probably came from Auschwitz. The experiments took place in Neuengamme. Later I saw a paper on these experiments which was written for Himmler. But it was written in such a scientific way that I did not understand a thing.The experiments upon and murders of the children are described in this pdf, taken from the book If you want to see your mother step forward - The 20 children of Bullenhuser Damm. A hug for the memory by Maria Pia Bernicchia. The photographs Heissmeyer took of the children he had operated upon were located in 1964 after Heissmeyer had told the Magdeburg prosecutors that he had buried a box containing his experimental records in his garden at home in Hohenlychen (Schwarberg, 1984, pp.108-109).
The books by Schwarberg and Bernicchia enable testimonies concerning the killings of the children to be collated. The following timeline is apparent (Schwarberg, pp.92-106):
May 15, 1945: Arrest of Pauly in Flensburg.
February 1, 1946: Arrest of Trzebinski
March 9, 1946: Depositions of Adolf Speck and Johann Frahm to investigating officer Captain H.P. Kinsleigh.
May 2, 1946: Detailed description by Frahm:
A rope was put around their necks and they were hung up on hooks like pictures on a wall
May 3, 1946: Pauly, Dreimann, Speck and Trzebinski were sentenced to death by Chief Justice C.L. Stirling
May 24, 1946: Trzebinski made a private note in his death cell:
I cannot reproach myself for giving the children a merciful injection of morphine before their execution. On the contrary, this was a humane act, of which I don't have to be ashamed.July 3rd, 1946: Deposition of Pauly (NO-1201)
October 2, 1946: The condemned men were driven to Hameln for execution. After his arrival, Trzebinski wrote a farewell letter to his wife:
But if, after all possible attempts to save the children and after finally realizing that they could not be saved, not by anyone - if after all this I gave the children a merciful weak injection of morphine, not to kill them but to ease their last hour before they were finally put to death at the hands of others, then this is a crime.October 8, 1946: The accused found guilty of the Neuengamme crimes were hanged.
We can therefore conclude that Pauly and Trzebinski continued to admit that these murders took place, even as they waited to be hanged with no hope of reprieve. In Pauly's case, the admissions included gassing and the transfer of some inmates for euthanasia "executions".