Is there an English equivalent of the Scots usage of "boak" (meaning retch) as a noun?

Puke is both verb and noun.

Cambridge

Puke:

noun - vomit

Verb - to vomit

Hence:
He puked after drinking too much and
he left a pool of puke on the ground.

Although it satisfies your title question, it does not work in the more restricted sense you mention of “gives me the boak”, although you might say “… gives me the vomits”, a phrase that is easily found in online search but interestingly is couched in the plural. Nevertheless, vomit is another candidate, being both verb and noun.

"Makes me gag" is a common expression in English that can be an equivalent of "that gies (gives) me the boak".

The expression has the verb gag meaning "to retch" (make the sound and movement of vomiting). There aren't any senses of noun gag that means vomiting or retching. However, there is gag reflex and you could say "gives me the gag reflex" which is used but not common.

I could smell Southern Comfort around me somewhere, a scent that almost immediately gives me the gag reflex.

_{Death Knell A John Keegan Mystery By John Misak}

The definition of gag reflex from OED:

n. retching, or spasmodic contraction of pharyngeal muscles, esp. triggered by touching structures in the back of the mouth; also figurative.

Additionally, retch is used both as a verb and a noun. The noun usage is not that common. Also, as you said, "gives me the retch" is not used. However, I was curious and I've searched the phrase in Google which only showed two results. (from sources that are not credible).

The definition of noun retch from OED:

An act of retching (in various senses); esp. an involuntary effort to vomit.

The verb retch can have two meanings in this context. The definitions of verb retch from OED:

2.

a. transitive. To expel from the stomach or oesophagus by vomiting. Frequently with up and out.

b. intransitive. To make an involuntary effort to vomit; to strain while making the sound or action of vomiting but without expelling anything from the stomach or oesophagus. Also occasionally: to expel matter by vomiting; to throw up.

I don't think there's a noun for this feeling. We normally say

That makes me nauseous.

Lexico defines it as

Affected with nausea; inclined to vomit.

The heaves is according to Collins a UK English phrase for "an attack of vomiting or retching". It must have some currency in the US as they quote American science fiction writer Sheri S. Tepper: "He slept until mid-afternoon and awoke without the heaves."

There are quite a few of such expressions in English for other sensations, but mainly for fear or nervousness. If you're afraid you may have the shakes, the heebie-jeebies, the willies, the wobblies, the jitters, etc.