Local Government Landfill Compliance Deadline Act - Exempts municipal solid waste landfills existing as of October 8, 1993, which meet criteria under the Solid Waste Disposal Act as in effect on January 1, 1993, from meeting any revised criteria (established for facilities which may receive hazardous wastes) for a period of one year. Makes such exemption inapplicable to landfill owners or operators who did not make a good faith effort before October 9, 1993, to comply with criteria. Provides for a 180-day extension of the exemption subject to specified factors beyond the control of the owner or operator. Directs the Administrator of the Environmental Protection Agency to issue regulations to revise specified financial assurance requirements under the Code of Federal Regulations (CFR) to broaden the mechanisms available to municipal landfill owners or operators for demonstrating financial responsibility. Makes current requirements inapplicable until the effective date of such regulations. Requires the Administrator to issue regulations that exempt certain small landfills from CFR groundwater monitoring requirements.