Sir Peter Westmacott became Britain’s ambassador to the United States in January, not long after the Conservative-led government undertook drastic budget cuts that helped push the British economy into a double-dip recession. Westmacott, who lives in Washington, visited Boston recently and spoke with Globe reporter ­Megan Woolhouse about the nation’s plan to attract more businesses to its shores and how the UK’s leaders are hopeful that US investors can help.