British trade to countries that have deals with the EU has increased dramatically, generating business worth billions of pounds, according to new research released by the main pro-EU campaign.

In a challenge to anti-EU campaigners who say that Britain could negotiate better deals outside the union, the Britain Stronger in Europe campaign says EU free trade agreements have led to a doubling of exports in some cases.

The pro-EU campaign is publishing a report, written by three former UK ambassadors to the EU, that argues British exporters would be disadvantaged if the UK left the union. Britain would immediately lose access to free trade deals with 51 states. It would then have to renegotiate its own bilateral deals as a weaker force – a country with a population of 63m rather than as a member of a union with a population of 500m.

Sir Nigel Sheinwald, one of the report’s authors, who served as the UK’s permanent representative to the EU in Brussels between 2000-03, told the Guardian: “Outside the EU we would be worse off. I don’t think any of us are saying we would never get a deal with these countries. I just think we would be at the back of the queue and it would take a long time.”

The report, also written by Lord Kerr of Kinlochard and Lord Hannay of Chiswick, found that:

UK exports to South Korea have increased by 104%, worth £3.7bn, since the EU-South Korea free trade agreement came into force in 2011. UK exports to Canada have increased by 29%, the equivalent of £2.3bn a year, since a free trade deal was signed between the EU and Canada in 2013.

No European country outside the EU has been able to negotiate a trade deal to give it access to the EU’s single market on the same, or better, basis as that enjoyed by EU members.

UK trade outside the EU is growing. But 45% of the UK’s goods and services exports still go to the EU compared with 8.5% to the emerging Bric countries of Brazil, Russia, India and China.

As EU countries sell only 6% of their goods to the UK, Britain would have a weaker hand in negotiating a trade deal with the EU as an outsider.

The former ambassadors also question the Eurosceptic claim that British membership of the EU hampers its ability to trade with non-EU countries. The ambassadors say the US has either been the largest or second largest destination for British goods exports in every decade since the 1960s – both before and after the UK joined the EEC in 1975. Germany, the EU’s largest economy, has no difficulty in trading outside the EU. Its exports to the Bric countries measured £90bn in 2013 – way ahead of the UK on £37.03bn.

The former ambassadors conclude: “In commercial relationships, there is strength and safety in numbers and we are much better off standing together with the rest of the EU than we would be on our own. It is false to claim that the UK has to choose between forging international trade agreements and playing an active part in EU trade policy.

“The UK’s membership of the EU has enabled it to leverage the greater weight of the EU in international trade negotiations to open up world trade, creating jobs and growth at home and bringing much-needed foreign investment into the UK.”

Sheinwald said anti-EU campaigners had failed to understand that the UK’s biggest trading partners, such as the US and China, expect to negotiate trade deals with regions rather than on a bilateral basis. He cited the recent remarks by Mike Froman, the US trade representative, who said last week: “We’re not particularly in the market for FTAs with individual countries. We’re building platforms … that other countries can join over time.”

Sheinwald said: “The way the world trade negotiations have developed over the past decade – from a universal-type UN negotiation in the WTO to more region-to-region, block-to-block type negotiation – will disadvantage a medium-sized country if it were to go it alone. Whereas at the moment the EU is able to do these big deals with Korea and we hope with the US. But by ourselves we would have to negotiate 50 or 60 individual trade deals. For the big countries like America their preoccupation is doing these regional ones, with Europe and obviously finalising the one getting through Congress with Asia.”

Sheinwald also said that the anti-EU campaigners had failed to answer the charge that Britain would have a weak hand in its exit negotiations in which two member states would negotiate on behalf of the remaining 27 members of the EU. Any deal would have to be agreed by the European parliament. “A little bit like Greece found during the summer, it will be the bloc versus the one. At the moment in our negotiation we are able to play as a member of the club, just about. But we will not be able to play as a member of the club because we will be on our way out of it at that point.”