(Updates with United States ISM data, Mexico PMI)
* Increasing factory activity carries into 2017
* Risks ahead from U.S. protectionism, Brexit, elections
* U.S. manufacturing growing at fastest rate since Nov 2014
* Euro zone factory activity at near six-year high
* China's activity expands for sixth month in a row
* Japanese manufacturing growth fastest in almost three years
* Graphic on Reuters economic polls tmsnrt.rs/2e7JFphttpt
By Jonathan Cable and Nichola Saminather
LONDON/SINGAPORE, Feb 1 (Reuters) - Factories across the world fired up - or at least kept up - activity in January with some registering multi-year output highs, just as a barrage of political risks threatens the global economy with potential harm.
Rising protectionism from the United States, concerns over how Britain's negotiations on leaving the European Union will pan out, and national elections in Europe's largest economies all lie ahead.
But entering 2017, economic growth gathered momentum, according to surveys released on Wednesday, following on from last year thanks to a bounce in consumption.
Manufacturers in the United States, the world's biggest economy, last month increased activity quicker than at any time since late-2014, took on far more workers than was expected and saw heightened demand for their goods.
Euro zone factories registered the fastest activity rate for nearly six years, China's activity expanded for the sixth month and Japanese manufacturing growth was the fastest in almost three years.
Even in Britain, where a slump in sterling since the June referendum stoked the sharpest rise in factory costs on record last month, growth remained robust.
"So far momentum is pretty strong heading into 2017," said Jacqui Douglas at TD Securities. "But political risks are definitely one of the biggest this year and given the surprises we had through 2016 it's really hard to tell what's in store."
Among unexpected events last year was Britain's vote to leave the EU and the election as U.S president of Donald Trump, both seen as the result of anti-establishment anger among voters who feel left out of the wealth of nations.
Signs of concern this may spread could be found on bond markets. The premium investors demand to hold France's government debt rather than of similar economies shot up on so-called Frexit fears - the possibility the far-right National Front might win the presidential election and try to take the country out of the euro zone.
IHS Markit's final manufacturing Purchasing Managers' Index for the currency bloc rose to 55.2 in January from December's 54.9, its highest since April 2011. A Markit/CIPS UK factory PMI edged down to 55.9 from December's 2-1/2 year peak of 56.1, matching the consensus forecast in a Reuters poll. Anything above 50 indicates growth.
The U.S. Institute for Supply Management said its index of national factory activity rose to 56.0 last month, the highest since November 2014. However, construction spending unexpectedly fell in December.
Across the border in Mexico, growth in manufacturing edged up in January from a more than three-year low. Uneven U.S. demand for Mexican factory exports has weighed on growth and Trump has added uncertainty by threatening tariffs on Mexican-made goods.
A stronger dollar helped economies such as Japan, where export orders surged, PMIs showed, a welcome sign alongside recent data suggesting a more durable recovery may be underway.
However, those encouraging signals sit uncomfortably with the threat from Trump's trade policies. Japan is moving to temper the risks with plans to show Trump its firms are ready to create U.S. jobs, according to a document whose contents were revealed to Reuters.
In export-reliant global supply chains are closely interlinked, and Trump's election is a particular risk to both world trade and broad economic growth if the new president follows though on his "America First" policies.
"The uncertainty surrounding future market access to the U.S. is bound to weigh on investment activity as companies await regulatory certainty," said Frederic Neumann at HSBC.
In China, the world's second-biggest economy, growth was led by an investment and construction boom that has helped spur global growth. Its official PMI was 51.3 in January, slowing marginally from 51.4.
Analysts question whether Chinese growth will be sustainable once the impact of earlier stimulus wears off and if the property market cools. They warn a slowdown in the economic powerhouse could ripple across the region and beyond.
"Within China, we expect that real estate will slow down, because the government is quite keen to contain housing prices," said Louis Kuijs at Oxford Economics.
Other regional economies like Indonesia showed positive momentum in manufacturing activity, while Indian factory activity returned to modest growth in January, bouncing from a contraction triggered by the government's scrapping of high value banknotes.
Even in laggard South Korea exports rose at the fastest pace in nearly five years.
"We remain quite cautious how much of an acceleration in growth we can see in this pretty challenging climate," Oxford Economics' Kuijs said.
"Things like PMI are timely indicators of the hard data but sometimes they do run ahead, and the improvement in actual data doesn't materialise."
Editing by Jeremy Gaunt