I personally wouldn't do HEX as I find Paddington inconvenient in every possible way. If it were me, I would have just taken the Piccadilly line (a tube is a train after all! ) to Leicester Square, concert at St Martin in the Fields (you are a stone throw away from the Royal Opera House but even a matinee concert would presumably take too much of your time in the after all), walk in St James Park and enjoy the views of Buckingham or the Queen's guard, then as suggested above transport museum at Covent Garden. The advantage of that programme is that all of this is very close together (Westminster is close too if you wanted it as well) so while it will take some walking, you'd do your entire day without needing any transport apart from to/from Heathrow. A tonne of great restaurants around Covent Garden and Soho too depending on what you fancy. For instance, if you fancy something very British, you could do Sunday roast at Hawksmoor in either Regent St (5 minutes walk from St James Park) or Seven Dials (5 minutes walk from Covent Garden. They are fairly pricey restaurants but strangely enough their Sunday roast is not. Some excellent choices for Indian, Chinese, Vietnamese, Japanese, etc in the area if you prefer something more exotic.