As far as I know, this scheme, like other visa-related bilateral agreements with Schengen countries, only applies to one country at a time (several other Schengen countries are also part of the working holiday scheme). What you will get is a type D visa that does allow you to travel elsewhere in the Schengen area, but not to exceed the three-months-in-any-six-month-period limit for all other Schengen countries taken together.
It makes sense: if one country could issue one-year visa for the whole area, it would defeat the whole system and give that one country exorbitant powers on the visa policy of its European partners. It's the same for residence permits, they can only be transferred from one country to the other after some time, under restrictive conditions (European blue card). Schengen countries generally remain free to set rules regarding long-term stay and immigration for non-EU nationals (except if they belong to the family of an EU national).
What you could in principle do is stay 1 year in the Netherlands and then another 90 days elsewhere in Schengen. In that case, it is advisable (necessary?) to leave the Schengen area and reenter (e.g. from the UK), to have a proper entry stamp and avoid trouble when leaving. Leaving from a Dutch airport, while by no means a requirement, might also make things easier.
In practice, you could also easily drive or take the train to Belgium, France, Germany, etc. as there are no systematic checks. The thing is that if you are staying in the Netherlands, you are already in the Schengen area and tracking the time you spend in other countries is more difficult as your passport won't be stamped every time you cross an internal Schengen border. So visiting from time to time and formally “overstaying” for a few days should be OK.
Flying sounds a little more risky but might still be OK, at least if you haven't used up your three months already (ground handling personnel will look quickly at your passport but there are no systematic police checks, no stamps and no look-up in Schengen databases).
If you have bad luck or get into trouble for some reason, the police can however still check your status, and not only at the border. So if you find yourself staying somewhere illegally, there is always a potential for problems and various kinds of unpleasantness.
Finally, do check well in advance that you will be able to apply for a work holiday visa from within the Netherlands. Some countries allow that but others don't and require you to apply from your country of origin or, at least, another country where you are residing legally.