Further evidence of a nationwide recession came yesterday with official reports of a 0.6 percent drop in productivity for all nonfinancial corporations in the third quarter, a 1.6 percent productivity decline among private firms excluding farms, and a 0.1 percent decline in real, pretax average earnings.

The personal tax cut that went into effect last month, however, helped the average worker stay just ahead of inflation in October, with real spendable earnings, after taxes, rising by 0.3 percent. The sharp slowdown in measured inflation in October also held up real earnings, which had dropped 1.7 percent in September.

Figures for this month are likely to show a further decline in spending power as inflation rebounds somewhat from the abnormally low rate in October.

Commerce Secretary Malcolm Baldrige acknowledged in a speech to businessmen Tuesday that the economy is in for "some very rough times" in the next few months. Private economists have also been revising their forecasts downward as the evidence of a recession builds up.

The administration is still predicting a strong economic recovery next year, but officials had suggested earlier that it would begin in the second quarter. Baldrige said that after "a very difficult fourth quarter this year and a tough first quarter next year" there will be a better second quarter and "we should see real recovery around mid-year."

He blamed the recession on President Carter's policies, but most analysts believe it was brought on by record high interest rates throughout this summer, which resulted from the Federal Reserve's tight money policy and financial market fears that the national budget deficit would increase under President Reagan's policies.

The October rise of spendable average earnings was the first increase of more than 0.05 percent in five months, but the spending power of the average worker was still 3.6 percent lower last month than a year earlier. Over the past year hourly earnings have risen by 8.2 percent on average, while prices have gone up 10.1 percent and the average tax burden has increased by more than 1 percent.

Workers in wholesale and retail trade have been affected most in the last year. Their average hourly earnings after inflation, but before accounting for tax changes, fell by 2.4 percent between October 1980 and last month. Construction workers saw a 2.3 percent cut in pre-tax living standards. Mining workers secured a 9.4 percent rise in average hourly earnings, limiting to 0.6 percent the decline of what can be purchased with an hour's pay.

Recession usually leads to a drop in productivity as firms cut their production faster than they lay off workers. Unemployment has now started to rise sharply, but not by enough to offset weaker output. Yesterday's official figures, however, showed a lesser decline in non-farm business productivity during the quarter than was initially reported. The decline is now put at 1.6 percent.

In manufacturing there was a 1.3 percent rise in productivity, with a 2.4 percent jump in nondurable goods and 0.5 percent increase in the durable-goods sector.

The overall productivity decline in nonfinancial corporations was the first for this sector since the second quarter of 1980. It reflected a 0.2 percent drop in output and a 0.4 percent increase in hours worked by all employes.

One beneficial side effect of the recession is that interest rates fall as private-sector credit demand declines. Most major banks brought their prime lending rates down to 16 percent this week. Chase Manhattan Bank went further, to 15 3/4 percent on Tuesday, and yesterday three banks in Racine, Wis., put their primes at 15 percent. The Heritage Bank and Trust, the Heritage Bank of Mount Pleasant and the Heritage New Bank are all in the same holding company in Racine.