(Adds reaction from economists)

By David Milliken

LONDON, Sept 3 (Reuters) - Businesses in Britain's services sector recorded their slowest growth in more than two years last month, mirroring signs of economic weakness in the United States and China, a closely watched survey showed on Thursday.

Financial data company Markit, which compiled the figures, said Britain's overall economic growth rate was likely to slow to 0.5 percent in the three months to September from an above-average 0.7 percent in the second quarter.

"Even after allowing for usual seasonal influences, August saw an unexpectedly sharp slowing in the pace of economic growth," Markit's chief economist Chris Williamson said.

The Markit/CIPS Purchasing Managers' Index (PMI) for Britain's services industry -- which does not cover retailers or the public sector -- dropped in August to its lowest since May 2013 at 55.6, down from 57.4 in July.

That was below all forecasts by economists in a Reuters poll that had predicted a modest rise, and sterling briefly fell.

BNP Paribas economist Dominic Bryant, who had predicted a small decline, said he expected the PMI to recover slightly in the next couple of months, but not to return its previous level.

"In 2016 ... a combination of fiscal tightening and modest global growth is likely to see growth slow further," he said.

Uncertainty about the scale of China's slowdown has caused share prices to slump in recent weeks, and has triggered broader worries about world growth which may delay the United States' first increase in interest rates since the financial crisis.

Many economists think the U.S. Federal Reserve could move as soon as next week, though figures on Tuesday showed the weakest growth for more than two years at Chinese and U.S. factories, as well as a softening in China's services sector.

The Bank of England is also considering when to start to raise interest rates although financial markets do not expect it to act until next year.

British inflation is barely above zero and Thursday's survey suggested it was unlikely to rise fast, with the second-smallest rise in input costs for British services companies since 2009.

"The inflation outlook is benign and is therefore likely to help tip the argument towards postponing any rate hikes until the wider global economic picture becomes clearer," Williamson said.

A broader Markit index -- which wraps in construction firms and factories -- also fell to its lowest since May 2013, and pointed to a return to more average annual rates of growth of around 2 percent after a rapid expansion in 2014 and early 2015.