How much is the USA dependent on British and French support in international relations?
While no nation has a closer alliance with the United States than the UK, France's history with the United States is different and a bit more contentious at times. The UK and France are among the United States most important allies.
Both are liberal democracies which generally have common concerns with the US on international issues.
Both are on the UN Security Council and that means nothing substantial goes through the UN without their agreement. Both have veto's on any substantive UN action.
Both are important NATO allies who drill, coordinate and consult with the US on defense. NATO is America's most important ally and Europe is a region which America twice has gone to war over (WWI, WWII).
Both along with the United State have the 3 most capable militaries able to project power removed from their borders more than any other nations including Russia and China. So If the US were to go to war removed from their common boarders no other two countries would have the ability to support the US militarily than the UK and France.
The UK and France along with the United States are among the top eight largest economies in the world, another important group for international relations and decision making.
The UK is the closest ally the United States has. It's among the five eyes allies made up of the UK, the US and some of the UK's commonwealth nations. This group by treaty shares a special relationship with regards to security assets including raw signals intelligence, technology and many sensitive defense secrets shared with no other US allies.
When Iraq invaded Kuwait ultimately resulting in the first gulf war. Then American President George H W Bush decided to go to war largely on the strong advice of British PM Margaret Thatcher. She was so respected among American political leadership at that time that her say so was enough to convince them to go to war. In interviews prior to his meeting with Thatcher President Bush and his advisors were not pursuing military action. It was Thatcher who convinced Bush that Saddam's aggression should not be permitted to stand. Appeasing aggression yields more aggression. I've read that from several different sources.
The JCPOA is a good example, even if three other countries were involved (China, Russia, Germany).
Working with Europe, China and Russia, the United States pressured Iran into a deal which froze their uranium enrichment in exchange for sanctions reduction. Various US agencies certified that Iran kept that narrow agreement, while working against Western interests in many other areas.
Later on, working against Europe, the United States tried to pressure Iran into going beyond the reductions agreed in the JCPOA. As a result, Iran is again enriching uranium beyond 3.67%.