When I used to deal with the tax affairs of Non Domiciled Residents 50 years ago, they would only pay UK tax on any foreign income if that money came into the UK. If the income was paid into another tax jurisdiction, they would be liable to pay tax under that country's rules.
How would you legislate for all world wide income, for all Non- Resident Domiciles to be taxed in the UK, including repealing the multitude of Double Taxation Treaties the UK has with most countries around the world?
Under the 1993 UK/India Double Taxation Treaty, the maximum tax deducted from dividend payments is 15%. Should the income tax charged be greater than 15% the recipient would be entitled to a tax credit for the difference over that amount.
Regarding any US income of Ed Balls, I would presume that US tax would have been deducted at source, then properly treated by the UK Inland Revenue under the UK/USA Double Taxation Treaty via his annual tax return.