By Jonathan Cable and Leika Kihara

LONDON/TOKYO (Reuters) - Global factory activity weakened in September as soft demand and economic uncertainty pointed to a tough outlook, surveys showed, keeping policymakers under pressure to shore up fragile growth.

Manufacturing activity across the euro zone slowed at its fastest pace this year as demand waned sharply despite factories cutting prices and Germany, Europe's largest economy, recorded its most pronounced worsening of conditions for 12 months.

HCOB's final euro zone manufacturing Purchasing Managers' Index (PMI), compiled by S&P Global, dropped to 45.0 in September. This was just ahead of a 44.8 preliminary estimate but further from the 50 mark separating growth from contraction.

"The euro zone rebound anticipated in early 2024 has turned out to be quite tepid. Confidence remains somewhat depressed and the manufacturing sector still looks very weak," said Natasha May at JP Morgan Asset Management.

A drop in oil prices helped bring down input costs in the region but there are worries escalating Middle East tensions could impact output and drive them back up.

European Central Bank President Christine Lagarde dropped the clearest hint yet about another interest rate cut on Monday while Federal Reserve Chair Jerome Powell indicated the U.S. central bank would likely stick with quarter-percentage-point interest rate cuts moving forward and was not "in a hurry".

Inflation in the 20-country currency union fell to 1.8% in September, below the ECB's 2% target and reinforcing an already solid case for another interest rate cut this month, official data showed on Tuesday.

In Britain, outside the European Union, factory managers turned sharply more pessimistic as worries about the new government's first budget combined with concerns about the Middle East and strong inflation pressures.

SLOWDOWN

Still, Asian manufacturers may get some relief in coming months from aggressive stimulus unveiled by Chinese authorities over the past week, including a lowering of interest rates and an injection of liquidity into the banking system.

But factory activity in Japan shrank in September and expanded at a slower pace in Taiwan, PMIs showed, highlighting the toll soft global demand was taking on Asian exporters.

In a sign of a widening fallout from slowing U.S. growth, South Korea's export growth decelerated in September with shipments to the world's largest economy barely increasing.

In China, factories struggled to make headway, with the Caixin/S&P Global manufacturing PMI on Monday showing a slump to 49.3 in September from 50.4, the lowest reading since July 2023.

It was a similar picture in Japan, which is relying on exports to boost economic growth amid subdued consumption. The final au Jibun Bank Japan PMI dipped to 49.7 in September from 49.8, below the 50.0 threshold for the third straight month.

"Softer growth in new orders was the main factor weighing on manufacturing last month," said Shivaan Tandon, markets economist at Capital Economics, on Asia's PMI.

"Global demand is set to remain weak in the coming months and weigh on activity in Asia for the near term," he said.

The PMI for Taiwan stood at 50.8 in September, falling from 51.5. Manufacturing activity shrank in Vietnam, Malaysia and Indonesia, the surveys showed.

Growth in India's manufacturing industry cooled to an eight-month low in September as new orders - a key gauge of demand - grew at the weakest pace since December.

The International Monetary Fund (IMF) anticipates a soft landing for Asia's economies as moderating inflation creates room for central banks to ease monetary policies. It predicts growth in the region to slow from 5% in 2023 to 4.5% this year and 4.3% in 2025.