By Caroline Copley

BERLIN (Reuters) - German engineers, key to the country's export-powered economy, expect uncertainty about Britain's future ties with the European Union to drag on exports but do not want future UK access to the EU's single market to come at any price.

Exports to Britain fell 4 percent in the first quarter in the run-up to the UK's June 23 vote to leave the EU, the VDMA engineering association said, and ensuing uncertainty is likely to further sap demand for vaunted "Made in Germany" goods.

"Companies are facing insecurity. What I am hearing now from companies I know is that some of them are trying to relocate their business from the UK to the continent," said Holger Kunze, director of the VDMA's European office in Brussels.

German firms exported goods worth 89 billion euros ($98.6 billion) to Britain in 2015, making the UK their third most important export destination after the United States and France.

Kunze said the VDMA would prefer a "Norwegian model" for future trade with Britain which would give the UK access to the single market in return for paying a contribution to the EU budget and guaranteeing freedom of movement for EU nationals.

One of the major reasons cited for Britain's vote to exit the EU was uncontrolled immigration from the rest of the bloc.

Nevertheless, Kunze told Reuters, the priority in future Brexit negotiations with London should be to ensure that the other 27 members of the EU stick together.

"The bottom line is, let's save the European Union, that's the first priority. Let's see that the UK stays as close as possible to the internal market, but this cannot be achieved at any price," Kunze said.

Some VDMA members are already concerned about barriers to free movement, Kunze said. One German firm which had recently concluded a contract to build a plant in Britain was worried about how its experts would be able to travel there to follow up on construction work in the event of Brexit.

Separately, Eric Schweitzer, president of the DIHK Chambers of Commerce and Industry, said companies wanted to scale back plans for investment and employment in Britain and feared barriers to trade.

"The political and legal uncertainty in the transition period is already leading to falling exports to Britain at one in four companies," he said in a statement.

On Thursday, the DIHK said it now sees exports to Britain to fall 1 percent this year, down from a previous forecast for a 5 percent rise. In 2017, it expects a 5 percent drop.

Schweitzer said he also believed it was important that negotiators found the right balance between good future trade relations with Britain and ensuring that the single market continued to function well.

"Brexit has made abundantly clear what an important achievement the integrated EU single market is for German companies. The EU should make every endeavor to ensure the cohesion of the 27 member states."

(Reporting by Caroline Copley; Editing by Mark Heinrich)