Which persons of the Trinity are attached to which noun in John 1:18?

It's important to note there are 2 major textual variants here. The other answer deals with the (easier to paraphrase into persons of the Trinity) variant. The questioner asks about the more difficult variant, and in particular how that variant is translated by the ESV.

The ESV translation

No one has ever seen God[1]; the only God[2], who[3] is at the Father's[4] side, he[5] has made him[6] known. (John 1:18 ESV)

The 'who is at the Father's side and has made him known' seems pretty straightforward. It is

'the Son is at the Father's side and the Son has made the Father known'.

So we have

'No one has ever seen God. The only God - the Son - is at the Father's side and the Son has made the Father know.'

The big problem here is that we now have Jesus being equated with the 'only God', but John 17:3 clearly shows the Father is the 'only God'.

Some hold that 'only true God' actually applies to any person, and is used to distinguish from false gods, such as in polytheistic systems.

Consider the comment here

https://hermeneutics.stackexchange.com/a/66900/36403

On this view, John 17:3 isn't meant to apply 'only true God' to the Father exclusively, but rather to God in general. John 17:3 could be paraphrased as "They may know the Father, who is a person of the only true God, and Jesus Christ, whom the Father has sent and who is another person of the only true God."

So then, similarly, 'only God' at John 1:18 refers to God in general. We need to paraphrase a bit, then.

'No one has ever seen God. The Son, who is a person of the only God, is at the Father's side and the Son has made the Father known.'

We now then must treat the first sentence. Since the last part is about making the Father known, and since Jesus has been seen by many people, this probably refers to the Father. So

'No one has ever seen the Father. The Son, who is a person of the only God, is at the Father's side and the Son has made the Father known.'

I leave it up to the reader whether this is satisfactory, given that 'only God' and the 'who' immediately following it in the ESV seem to indicate a tighter relationship than this. YMMV.

(We are left with another problem, which is the Son is at the Father's side but no one has ever seen the Father? We could paraphrase 'one' as 'man', and hold that although Jesus is man, he is not 'a man'. Or, we could infer an 'except' here. See below.)

NET translation

The NET translation is a bit easier to paraphrase, in my opinion.

"No one has ever seen God. The only one, himself God, who is in closest fellowship with the Father, has made God known."

It could be paraphrased as

"No one has ever seen the Father except Jesus, the only one who has seen the the Father, who himself is of the substance of God and who is in the closest fellowship with the Father, and has made the Father known."

It might seem that the first sentence is meant to mean no one at all, but see John 6:46, which is

"not that anyone has seen the Father except the One who is from God; only He has seen the Father."

But perhaps the 'only' is meant to apply to fellowship. In which case

"No one has ever seen the Father. Jesus, the only one who is in the closest fellowship with the Father, who himself is of the substance of God - this Jesus has made the Father known."

In this case, though, what of the Holy Spirit? Is the HS not in closest fellowship with the Father? So this seems problematic.

So a paraphrase of the textual variant the ESV relies on, using the NET translation, could be

"No one has ever seen the Father[1] except Jesus[3 and 5], the only one who has seen the Father, who himself is of the substance of God[2] and who is in the closest fellowship with the Father[4] - this Jesus has made the Father[6] known."

Having said this, my guess is the 'only' here is actually supposed to refer to Jesus being the only begotten (which doesn't seem to be how the ESV or NET translators are intending it) and so a 'Son' should be added as inferred, such as in the Good News Translation.

The Good News Translation

"No one has ever seen God. The only Son, who is the same as God and is at the Father's side, he has made him known."

This can be paraphrased as "No one has ever seen [the Father]. The only [Son], [this Son who] is the same as God and is at the Father's side, [the Son] has made [the Father] known."

We still have the tricky phrase "is the same as God". What does this mean? It could be clarified by adding something like

"No one has ever seen [the Father]. The only [Son], [this Son who] is of the same substance of God as the Father, and is at the Father's side, [this Son] has made [the Father] known."