Any municipality complying with the provisions of this chapter shall, at its annual meeting or at a special meeting thereof warned and held for such purpose, fix the date of a special meeting to be held prior to the beginning of the next ensuing fiscal year for the purpose of acting upon its budget, which meeting shall be designated as the annual budget meeting of such municipality. The budget-making authority of such municipality shall thereafter submit any estimates and recommendations required by § 7-344, § 12-122 or any provisions of special acts to the annual budget meeting rather than to the annual town meeting. The budget-making authority and the legislative body of any complying municipality having a board of finance created under the general statutes shall thereafter exercise the powers and discharge the duties imposed upon them under § 7-344, including the holding of the public hearing required therein, in connection with the annual budget meeting rather than the annual town meeting as prescribed in said section. The budget-making authority, the legislative body and the officials and departments of any complying municipality having a board of finance created under a special act shall thereafter severally exercise the powers and discharge the duties imposed upon them under such special act, including the holding of public hearings required therein, in connection with the annual budget meeting rather than the annual town meeting and upon dates set forth in the vote in compliance with the provisions of this chapter, such dates having the same relation to the annual budget meeting as the dates fixed in such special act have to the annual town meeting, with like force and effect as though fixed by the special act.