Not a big issue - now that Mysteryman has put you right Regular use of the machine should keep the pipes relatively clean . Wouldn`t hurt to keep the sink waste pipe clean and definitely don`t put fat down it - all pretty basic stuff , clean pipes flow freely - gunked ones are always compromisedAnd/or, should the trap for the washing machine be raised so that there is an incline from the sink ? Currently, it's level / slight decline.
Need to get this sorted asap.
Many thanks