One possibility might be to charge capital gains whenever shares are traded. Then require shares to only be traded at exchanges with an agreement to report trades (or anonymously collect capital gains and send them your way, for privacy).
Another (probably worse) alternative might be not to tax them. The US will, and in exchange you get to tax UK shareholders of US companies.
You don't. But then again why would you - that shareholder gets no services from the UK government so there's no moral case for them to pay tax to a foreign country.