As valuable as any Tier 1 wizard: wizards are generalists by design
The limit of the scope of this question is from levels 1-4. The wizard does not choose a school until level 2. A focus on levels 2-4 leads us to initial spell selection. That single feature, the six initial spells and three cantrips, selected before adventure day one, is step one in the Necromancer Wizard, or any wizard, being good for the party. The choices made for second level spells (a total of four of them at levels 3 and 4) are the next significant levels of being beneficial to the party.
There is no requirement to choose necromancy school spells initially, though choosing some is valid if the character is being built with an eye to the future. If you require that the character choose necromancy spells as they level up, as the DM, the problem isn't with the Necromancy school of wizardry. As a game design matter, levels 1-4 are not that crucial to character effectiveness (see PHB p. 15). A power and capability spike happens at level 5: multi-attacks for martial characters and 3rd level spells for casters. New features for all wizard sub classes arrive at level 6.
- Consider the necromancy school spell: Cause Fear (XGtE, p. 151).
You can take a monster and neutralize it for an encounter if it does
not make the save. (Monsters sometimes make saves). At 3rd level
that spell gets inside of the heads of two monsters. The
frightened condition helps the party; keyword crowd control.
- Shape the battlefield: Fog Cloud, level 1. (I'd consider that at level 2 for a pick). For part of the battle, some of the enemy can't see your party which provides a variety of tactical benefits. Later in the character's life, having the zombies or skeletons emerging from your own fog cloud is a neat cinematic effect.
All of the party's characters at levels 1-4 (Tier 1)...
... are effectively apprentice adventurers. (PHB, p. 15, Tiers of Play)
A core benefit at second level regarding necromancy school spells is that spells that are found cost half as much to put into the spell book -- as with each other school.
The other benefit, the occasional hit point gain for killing an enemy, takes pressure off of the healing resources of the whole party. Design paradigm: it's a team game.
- This is useful. Healing magic and Hit Dice for HP recovery are in
short supply at low levels. If the two burning hands spell takes out three goblins, or if two shatter spells takes out some goblins, that is 2 or 4 HP healed without expending a valuable HD, potion, or other healing resource. With arcane recovery at third level and 4th level the wizard can get a second level slot back, or two firsts, which may increase this benefit depending on which spells you choose to recover.
At first level
The problem is that there are too many good spell choices. The quick build offers an OK starting point for any school of magic.
... Second, choose the Sage background. Third, choose the light, mage hand, and ray of frost cantrips, along with the following 1st-level spells for your spellbook: burning hands, charm person, mage armor, magic missile, shield, and sleep. (Basic Rules page 30)
Experience base: six wizards played (Five, if the UA Loremaster is discounted, which IMO it should be).
I'd suggest firebolt over light depending on party make up, and chill touch over ray of frost since it has a chance to foil attacks by undead against the wizard casting it. (Also thematic for your necromancer). As I've not played with toll the dead I won't comment on it, but it looks to be a good fit.
- We discovered that chill touch is very handy if there are lots of undead in
the campaign. I was surprised enough at its usefulness when our
Eldritch Knight used it in one group that I made sure to choose it on
the next wizard I had the chance to play. The undead having disadvantage on an attack against your wizard is almost as good as a shield spell (which gives +5 AC) against that foe without it costing a spell slot. It also reduces the chance of a critical hit on you from 1/20 to 1/400; at low levels, crits can mean game over.
You can pick up area of effect spells, like shatter at level 3, and more battlefield control spells (example: web). The point is to help the team succeed. That is true for any wizard. Both of my brother's wizards (illusionist and transmutation schools) focused on party support and various attempts a battle field control at low level since that was the best way to leverage spell casting (beyond the use of cantrips).
There is no need to worry about the necromancer
I don't find the concern to have much merit, unless the game will only last until level 4. From the comment you provided, the campaign is expected to reach level 7. The Necromancer has plenty of chances to shine as new spells and class features arrive.
- Key Point About Wizards
They can theoretically find and place into their spell book every wizard spell in the game. No other arcane spell caster can do that.
- What I have found is that the tyranny of choice -- what do I prepare
today? What spells do I choose at this level? -- is as perplexing (if not moreso) as it was for the cleric I first played. Guessing at what we'd need beyond a few standard spells was an inexact science. This was true regardless of school. Evoker, Transmuter, Illusionist, Abjurer -- it didn't matter. (None of these got above level 7; campaigns dormant or dead).
How many additional spells the wizard has access to (beyond what they can pick at level increase) is up to you as the DM / adventure-loot quality control specialist. How many spells, spell books, or scrolls do you intend for them to discover at early levels? They are generalists as a class, regardless of magic school specialty.