There are basically two types of circuit breaker on a consumer unit (fuse board).
An RCD usually protects the whole house from Earth Leakage a potentially unsafe condition and when it trips both lights and power are lost until it's reset.
A Miniature Circuit Breaker (MCB) just protects one specific circuit - Uptairs Lights or Downstairs Power for instance. When an MCB trips it's because too much current is being drawn.
Your washing machine appears to be tripping an MCB because of an overcurrent situation - and as you say it happens almost immediately the machine begins to operate it may well be the water inlet solenoid (an easy fix).
John below may be almost right about the Heating Element but an open circuit anything will not trip a breaker - it may be a short circuit element and you may be able to eliminate the element by selecting a cold water wash if you are able to - no trip = probably the element.
The qualified electrician you spoke to either failed to understand what you were telling him or he has forgotten some of his basic training... Neither a Fuse nor the Earth would cause a breaker to trip.
Good luck...
Peccavi
May 2009