Undoubtedly the best way to get to Wednesbury is the M6. While it's not that far from Birmingham, it's a horrible journey by bike, to the point that I tend to limit trips to the Decathlon there to when I have use of a car.
Cheapest way from London would with with a West Midlands Trains advanced ticket via Birmingham New Street, and then either change to the Walsall line (there are a couple of stations in the WS10 area), or use the Metro (tram), depending on which bit of Wednesbury you're aiming for. If you need a walk-on ticket to Birmingham, the Chiltern line from Marylebone to Birmingham Moor Street is usually the better bet.