After several years of running a nursery business, I would say there aren't really any "must haves" other than good seeds. My top 3 would be sifted compost for germinating seeds, a good watering can that pours just how you like, and mulch for your nursery beds.
When propagating via rooted cuttings, a "propagation timer" automatic timer valve for running misting sprinklers 10 seconds every 10 minutes is sort of a "must have"
Not "must haves" but definitely makes life easier:
72-cell seedling flats (~3 cubic inches per cell)
Clear plastic humidity covers for germinating surface sown or light-dependent-germinating seeds (can also use a clear plastic bag with vent holes cut in it)
Refrigerator for cold stratifying seeds without freezing them
Moss as stratifying medium for medium to large seeds and nuts
Wood chips, pine needles, or leaf mulch for nursery beds
One of those long skinny shovels (a "drain spade") for digging out 2/3 year old seedlings, and a skinny hoe for scraping weeds in between
trees in your nursery beds.
If you have crows, ravens, jays, or other birds that like to pick out your freshly planted seeds, some kind of mesh cover or shade cloth draped over your beds will keep the birds from snacking on your seeds. If you have gophers,
voles, or
mice, a clove of garlic planted with every seedling will keep them from chomping your baby tree seedlings, or in the case of gophers will keep them from chewing on the the
roots from underneath.