The province of Hanover adopted several flags on the 5th December 1945:
The colours of the province were yellow-white, which means that the flag of the province was certainly yellow over white horizontally, exactly like the flag adopted for the Prussian province of Hanover in 1882. This time however (1945), Hanover was again a sovereign German state. The flag had certainly the proportions 3:5.
The provincial flag for the authorities is the one that Jan Oskar Engene mentioned. The proportions were 3:5 and the proportion between the height of the horse on the flag and the height of the flag was also 3:5. This flag was to be only on the buildings of the supreme president [Oberpräsident], of the president of the government and of each councillor and each mayor.
The vehicles of the authorities also displayed flags:
the provincial flag for the authorities but with yellow-white border, in the size 25 x 25 cm, for the supreme president.
the provincial flag for the authorities in the size 20 x 30 cm, for the president of the government
the provincial flag for the authorities in the form of a Wimpel (triangular flag), in the size 20 x 30 cm, for the councillors and the mayors.
All these flags disappeared when the province of Hanover merged with the Freistaaten [republics] of Brunswick, Schaumburg-Lippe, Oldenburg to become the new Land of Lower Saxony, but the law about the coat of arms, flag and seal of this Land (1952) re-officialised the flag of the province of Hanover as well as the other flags of the free states as regional flags. Source: Mattern and Neubecker 198x, II. Teil: Küstenländer. 2. Die Flächenstaaten in Zum Kleeblatt 1980-1981, pp. 98-100.
Pascal Vagnat, 26 May 1999
[According to the website of the Hanoverian monarchists] after the German defeat in 1945, the Prime Minister of Hanover Kopf (re)adopted the white horse on red in August 1946. After the founding of Lower Saxony late in November 1946, apparently a yellow-white flag with the uncrowned coat-of-arms (like the Hanover Province Official Flag in FOTW) was used until 1952.
Norman Martin, 26 June 2000
The coat of arms of the former province and Land of Hanover was Gules a horse Argent. It can still be used.
Pascal Vagnat, 26 May 1999
Provinzialdienstflagge / Flag for Provincial Authorities
image by M. Schmöger, 6 Oct 2000
In Veddeler 1996 there is information on the Provinzialdienstflagge of the province of Hannover, introduced in late 1945. This flag was simply a white horse on a red field. According to Veddeler's extraordinarily well documented work (see page 99-100 for a colour photograph and discussion and page 156 for reference to the relevant documents), the flag was adopted by the Oberpräsident for use on public buildings. The adoption date of the Verordnung by which the flag was adopted was 5 December 1945.
Jan Oskar Engene, 20 May 1999
Hinrich Kopf was Regierungspräsident and Oberpräsident of Hanover 1945-1946, and Ministerpräsident 1946-1955 of Lower Saxony. (...) The flag is actually the same.
M. Schmöger, 6 Oct 2000