Prospects for initial operation of a cable TV system in Fairfax County by late next year or early 1983 brightened considerably yesterday when the county's supervision rejected further study of the issue and directed an immediate start on the long-delayed franchising process.

"We've dickered around with this thing too long," Supervisor Thomas M. Davis III said as he offered a motion ordering the county staff to draft by mid-September a "request for proposals" from about 20 firms that have been jockeying for the potentially lucrative contract. Davis' proposal won unanimous approval from the board.

Under the board's timetable, a cable TV franchise could be awarded by next April or May, according to Fairfax Consumer Affairs Director Ronald B. Mallard. "And," he said, "it is reasonable to expect that the first part of the system will be operational within six to nine months."

If that happens, as many as 20,000 Fairfax homes could be wired by then for a cable system that might offer up to 108 channels and a host of options, such as first-run movies, two-way communications and home-security systems.

Under the county's cable ordinance, the winning firm will have up to five years to wire the county. According to staff estimates, about 50 percent of the county's 200,000 households would subscribe to the pay TV system.

Just how lucrative a franchise would be was underscored by a consultant's report received by the supervisors yesterday. It estimated that a cable system with 45 percent of all households would be worth from $160 million to $260 million in 15 years. The exact value would depend on how elaborate the system was.

Yesterday's decision to move ahead probably dealt a fatal blow to the hopes of some that there could be semipublic cooperative ownership. Supervisor James M. Scott (D-Providence) tried to win more study of cooperative ownership, but the board defeated that effort, 5 to 4.

Under cooperative ownership, citizens who were members of the cooperative would have a direct voice in what services would be offered, and in programming budgets and the rates for subscribers.

The key vote was cast by Supervisor Marie B. Travesky (R-Springfield), who said, "If a group wants to put itself together as a cooperative, that's fine. But I don't think we ought to spend any more time or money on the idea."

Despite his defeat, Scott said his idea "is not totally dead." He said private companies, in an effort to win board support, could offer at least partial ownership in their proposals.

The supervisors had already rejected the idea of public ownership, and though that option ws explored in the report made yesterday by CTIC Associates of Washington, it received no sympathy from the board."Municipal ownership is out of the question," said Supervisor Joseph Alexander (D-Lee), who last year saw county-owned cable as a potential boon that could provide tax relief for hard-pressed homeowners.

The consultant's report said that a basic system would require from $78 million to $94 million in capital expenditures, depending on how many homes were wired. The total costs for a more elaborate system could range from $135 million to $156 million, depending on the number of homes, the report estimated.

In order to achieve what it calls an acceptable rate of return -- 16.4 percent -- the report said the company that wins the Fairfax franchise will, after 15 years, have to charge subscribers from $33 to $59 monthly depending on the number of subscribers and how elaborate a system is installed. The monthly charges were adjusted for inflation.

After the board finished its three-hour discussion of cable TV yesterday, Chairman John F. Herrity, who has fought every attempt for additional study, said, "The board has been dealing with this for three years. aThe longer we delay, the higher the cost will be to the consumer. Now we've cleared the last of the roadblocks."

But Supervisor Martha V. Pennino (D-Centreville), whose original request for a delay led to an extensive study of public ownership and the implications of a fully wired county, said, "I am deeply grateful on bended knee that I asked for that delay."