Just on the football front the end of April is difficult to plan for if thinking of a top Premier League match as the TV companies will not have picked their live matches yet and any could shift from Saturday to Sunday or Monday (or even to a different time on the Ssturday).
Not to mention how difficult it is to get tickets, you'll probably have to commit before knowing the actual match day. Also premier league football in London is IMO ridiculously overpriced.
In the 3 lower leagues you have plenty of local London options West Ham, Millwall, Watford and Crystal Palace in the Championship - Brentford in League 1 and Barnet in Lge 2.
If Witshire or Oxford then Swindon Town, Oxford United and Wycombe Wanderers are at Home - Yeovil Town a bit further but not too far from Wilts.
Better value at those I'd say.