Do you mean Charing Cross? The regular trains from Ashford do go into Charing Cross, but there's also a new high-speed service to St Pancras (near King's Cross). If you get that, there are two direct buses (59 and 91 IIRC) from Euston Road opposite the stations.

Covent Garden is a 5-10 minute walk from the Waldorf, but if you were getting the Tube from there, I'd say walk to Holborn instead - a lot quieter on the weekend, and easier to find.