The country will begin it biennial shakeup of Congress Tuesday in Illinois, where Republican hopes for an auspicious takeoff in the first round of congressional primaries may hinge on the votes of no more than 12 people.

They are the jurors in the federal income-tax-evasion trial of Illinois Attorney General William J. Scott, who was the GOP's strong favorite to claim the seat of retiring Democratic Sen. Adlai E. Stevenson until Scott was indicted on charges of diverting campaign money to personal use and failing to report it for tax purposes.

Scott's once-wide lead in the three-way race for the Republican senatorial nomination all but vanished as the two-month trial not only kept him from campaigning but also dredged up even more allegations, including charges of double billing the state for travel expenses.

Polls now show Scott narrowly ahead of Lieutenant Governor Dave O'Neal, the most conservative of the three Republican contenders, with Peoria Mayor Richard E. Caver still far behind but holding out hope of profiting from the defection of moderates from Scott's camp.

By comparision, the Democrats are staging a tame show. Polls indicate that Illinois Secretary of State Alan J. Dixon is running far ahead of four other candidates, including Alex R. Seith, who ran against Republican Sen. Charles H. Percy in 1978 and at one point seemed to be on the verge of unseating him.

Although it appeared for a time that Scott's slow-moving defense in the courtroom would carry the trial beyond primary election day, the polls left little doubt that Scott might lose if the trial were still going on when the polls opened.

Scott rested his case Tuesday after disclosing late last week that he would not take the stand in his own defense. "It is imperative," he told reporters, "that a verdict be returned before the primary."

Any verdict by Tuesday would eliminate the possibility of Scott's being convicted after winning his party's senatorial nomination, which is a tantalizing prospect for the Democrats. A post-primary conviction would put pressure on Scott to step down from the ticket.

The conventional wisdom is that Scott will win the primary if acquitted in advance and lose it if convicted. But O'Neal strategists contend that the trial has been damaging enough to undermine the political advantage of an acquittal.

A Chicago Tribune poll indicated that while Scott was leading O'Neal 27 to 23 percent among Republicans, 37 percent of those surveyed were undecided and 60 percent said they might change their minds before election day.

The widely held view before the trial was that the Democrats had the most to fear from Scott, who was elected to his fourth term as state attorney general in 1978 with nearly two-thirds of the vote and who has a consumer and environment record to go with his generally conservative political background.

O'Neal, one Democratic strategist observed, would have a narrower base of support for the general election campaign.

As for the Senate, Stevenson and Abraham Ribicoff (Conn.) are the only incumbent Democrats stepping down this year. Three Republicans, Milton R. Young (N.D.), Richard S. Schweiker (Pa.), and Henry Bellmon (Okla.) are retiring.

Of the 34 Senate seats up this year, 24 are held by Democrats who were elected or reelected in 1974 in the aftermath of Watergate.

Republicans are counting on picking up three to six senate seats. The Democrats concede that several of their people are in trouble, including John Culver (Iowa), George McGovern (S.D.), Mike Gravel (Alaska) Frank Church (Idaho) and possibly, Partick J. Leahy (Vt.). The Republicans also list Birch Bayh (Ind.), John A. Durkin (N.H.) and Gary Hart (Colo.) among the most vulnerable Democrats.

With the Republicans now holding 41 of the 100 Senate seats, they are unlikely to take control of the Senate next year, but could position themselves within reach of that goal by the 1982 elections.

In the House, Republicans have 159 of 435 seats. Rep. Guy Vander Jagt (Mich.), chairman of the Republican Congressional Campaign Committee, says the GOP will do better this year than its 16-seat gain in 1978 and subsequent special elections.