La forme la plus proche, et la plus employée est:

Nous y sommes presque.

La forme suivante est équivalente, certes plus littérale, mais moins employée, car le pronom y est le plus souvent employé pour remplacer là:

Nous sommes presque là.

What is the best way to say, “we are nearly there”? Is it correct to say, “New sommes y presque”, or is, “nous sommes presque là”, better, or?

As Guillaume pointed in his answer, one way to say it would be

Nous y sommes presque. / Nous sommes presque là.

If your sentence’s context contain some temporal event, you could also say something like :

Nous y sommes bientôt. / Nous sommes bientôt là.

Another way to say it, would be using the ‘on’ pronoun.
If the context allows it, you could even say something like :

On arrive !

Quasiment plus personne ne dit « Nous y sommes presque » en dehors de contextes formels. Dans l’écrasante majorité des cas, We are almost there se traduit à l’oral par :

On y est presque. (un objectif)

ou

On est presque arrivés. (un trajet)

Almost nobody say Nous y sommes presque these days, outside formal contexts. In most cases, “We are almost there” can be translated in spoken French by:

On y est presque. (a goal)

or

On est presque arrivés. (a trip)

Note: Nous is essentially used by foreign people learning French these days. While French people are often eager to correct foreigners trying to speak French, they are never taught about this specific usage as it is not technically a mistake.

“Nous y sommes presque” is correct, and it’s idiomatic if the goal is an abstract one, i.e. when the sentence means “we’ve almost achieved our objective”. It isn’t very idiomatic (possible, but not common) to express that a journey is almost over. For that, we’d say “nous sommes presque arrivés”.

“Nous sommes presque là” doesn’t work. I can’t really explain why; là is a pronoun that means a place that’s somewhat distant, and it could fit in context, but the sentence just doesn’t sound good. Instead you need to use the pronoun y. Normally y refers to a place that has already been mentioned, but in the sentence “nous y sommes (presque)”, it can stand for an implied goal even if that goal hasn’t been mentioned explicitly.

In informal, spoken French, you can use “on y est presque” or “on est presque arrivés”.