Given the current state of affairs, your wife's status -- and yours -- will depend on when the UK leaves the EU, and whether it has done so with a "deal," that is, with a withdrawal agreement in place. If there is such an agreement, the deadline for moving to the UK is expected to be the the end of 2020.
If the UK is still in the EU when you arrive there, or if there is a transition period under the terms of a withdrawal agreement, your wife will be able to reside in the UK under the terms of EU free movement, allowing her to work and study in the UK. If she does so, you will not need a work visa to take up employment. In that case, both of you will be also able to apply for pre-settled status.
If you arrive in the UK after it has left the EU, and if there was no withdrawal agreement in place, then your wife may not be able to remain in the UK unless you manage to get her a dependent visa such as the Tier 2 family member visa. This carries with it a right to work.
Your wife will in any event be able to use the national health system; the only question is whether you'll have to pay the £400 yearly charge.
As noted in a comment, the situation is very volatile, and the state of affairs may have changed by the time you anticipate arriving in the UK.