As Dzug2 states, the time allowed for getting across London is usually extremely generous. (National Rail's system allows a minimum of one hour for all cross-London journeys, even if it's between King's Cross and St Pancras, which are only 2 minutes walk from each other!).
If you're travelling from Southend Central into Fenchurch Street, you've got about a 5 minute walk to Tower Hill, then about 15 minutes on the Circle Line to Euston Square and a 5 minute walk to Euston. If you're travelling from Southend Victoria into Liverpool Street you'll have a 10 minute trip on the Circle Line to Euston Square, plus a 5 minute walk to Euston.
If your train from Southend is cancelled, the station supervisor will validate your ticket for later travel and, if necessary, contact Virgin Rail's Control Centre to ensure that you don't have any problems using a later service out of Euston.
Chris