My honest answers as follows:
a) Is there a maximum time limit, within which the Police is obliged to complete their investigations, and submit their findings to the Court? If yes, what is this time period??
Ans: As per section 468 of Cr.P.C. no limit is prescribed for taking cognizance by the Court that means there is no prescribed limitation to file charge sheet in respect of offences which are punishable for more than 3 years.
b) How soon will the court take up the case , for hearing and judgement??
Ans. Immediately Court will take up the matter and usually, subject to availability witnesses, volume of concerned Court, it will be disposed between 2 to 4 years.
The relevant section is reproduced below. The option left with you for early filing of charge sheet and early disposal it to approach the Hon’ble HC and seek a direction to that effect, it will fix a time limit.
Section 468 in The Code Of Criminal Procedure, 1973
468. Bar to taking cognizance after lapse of the period of limitation.
(1) Except as otherwise provided elsewhere in this Code, no Court shall take cognizance of an offence of the category specified in sub- section (2), after the expiry of the period of limitation.
(2) The period of limitation shall be-
(a) six months, if the offence is punishable with fine only
1. Provisions of this Chapter shall not apply to certain economic offences, see the Economic Offences (Inapplicability of Limitation) Act, 1974 (12 of 1974 ), s. 2 end Sch.
(b) one year, if the offence is punishable with imprisonment for a term not exceeding one year;
(c) three years, if the offence is punishable with imprisonment for term exceeding one year but not exceeding three years.
(3) 1 For the purposes of this section, the period of limitation in relation to offences which may be tried together, shall be determined with reference to the offence which is punishable with the more severe punishment or, as the case may be, the most severe punishment.]