West German Chancellor Helmut Schmidt's Social Democratic Party suffered substantial losses in local elections in the state of Lower Saxony yesterday while the more conservative Christian Democrat Union gained strength.

The election results for city and district councils, reported today, confirmed a national trend of growing public disillusionment with the internal bickering over West German security and budget policies that has gone on for much of the year within Schmidt's party.

The setback came on top of a direct afront to Schmidt by a branch of his own party in the state of Schleswig-Holstein, which voted late yesterday against Western alliance plans to deploy new U.S. medium-range nuclear missiles in Western Europe starting late in 1983. It was the second time this year that a Social Democratic group at state level had opposed the missiles plan. The party in Baden-Wuerttemberg took a similar action in May.

Looking to next April's national party congress, the action carried ominous indications for Schmidt, who has threatened to resign if his party overturns its 1979 commitment to support the NATO policy.

A good deal of the unhappiness at local party levels with the way senior Social Democrats are governing in Bonn stems not only from opposition to the government's firm support of the missiles decision but also from displeasure with the budget austerity measures agreed to in negotiations with the Free Democratic Party, the Social Democrats' more free market-oriented coalition partner.

Showing no aversion at least to playing with the idea of a fall from power, Guenther Jansen, chairman of the Schleswig-Holstein party, told yesterday's party meeting, "the current social-liberal coalition is not the only way the Social Democrats . . . can make politics."

On the missiles, Jansen accused Schmidt's senior spokesmen of practicing increasingly "open, anti-Social Democratic propaganda." Willy Geusendam, a member of the state's Sosial Democratic executive committee, justified opposition to the NATO plan on grounds that "we are moving in the direction of war."

Meanwhile, the Social Democrats' election defeats in the neighboring northern state of Lower Saxony -- the fourth largest of 11 states -- indicated that while party officials continue to quarrel among themselves, Social Democratic voter ranks are thinning.

The party lost its absolute majority in several of the state's larger cities and districts. Overall, the Social Democrats slipped by more than six percentage points to 37 percent of the total, while the Christian Democratic Union gained 2.2 percentage points, rising to just over 50 percent. The total for the Free Democrats stayed roughly unchanged at 6.3 percent.

The Christian Democrats' national chairman, Helmut Kohl, hailed the outcome as a trend-setting rejection of Social Democratic rule. The Social Democrats' national manager, Peter Glotz, acknowledged that the losses had been influenced by Bonn politics and spoke of a danger to his party at two levels.

Glotz feared both that young voters would abandon the Social Democrats for the Greens and more left-wing alternatives, while working class voters -- the traditional backbone of the party -- would simply abstain out of confusion about where the Social Democratic Party is going.