Collins hat:

to frazzle

2. (fig: = tire) völlig erschöpfen or ermüden

 his nerves were frazzled - er war mit den Nerven völlig am Ende (inf)

M-W:

frazzle –

(trans.)

1: FRAY entry 2

2 a: to put in a state of extreme physical or nervous fatigue

b: UPSET

(intrans.)

1: the state of being frazzled

2: a condition of fatigue or nervous exhaustion

worn to a frazzle

https://www.merriam-webster.com/dictionary/frazzle

AHD:

frazzle –

(v. tr.)

1. To wear away along the edges; fray.

2. To exhaust physically or emotionally.

(v. intr.)

1. To become worn away along the edges.

2. To become exhausted physically or emotionally.

(n.)

1. A frayed or tattered condition.

2. A condition of exhaustion: worked themselves to a frazzle.

[Perhaps a blend of FRAY2 and dialectal fazzle, to unravel (from Middle English facelyn, to fray, from fasel, frayed edge, probably diminutive of fas, rootlets, from Old English fæs ).]

https://www.ahdictionary.com/word/search.html...

Random House unabridged:

frazzle – (Informal.) —v.i., v.t. 1. to fray; wear to threads or shreds. 2. to weary; tire out: Those six eight-year-olds frazzled me. —n. 3. state of being frazzled or worn-out. 4. a remnant; shred.

LEO:

frayed nerves – angespannte Nerven

Siehe Wörterbuch: frayed

frazzle – der Fetzen 

to frazzle (sth.) – (etw.[Akk.]) ausfransen

to frazzle – zerfetzen

frazzled Adj. – erschöpft

frazzled Adj. – entnervt

frazzled Adj. – fix und fertig (mit den Nerven)

frazzled Adj. – mit den Nerven runter

Siehe Wörterbuch: frazzle*

The Oxford and Macmillan senses (2) do seem like outliers, so if those aren't wrong, there may indeed be AE/BE differences.

>> ... 2 cause to shrivel up with burning:

'we frazzle our hair with heated appliances'

>> ... 2 [British] burnt

I'm not aware of any AE sense that involves burning, and no AE dictionaries mention it. For burning hair I think we would normally say 'singe' or 'frizz(le).'

My impression is that the sense of literally fraying, as of fabric, has mostly disappeared, but that frazzled nerves are simply another way of saying frayed nerves. 

That is, the emphasis is on mental stress and nervous tension, not on tiredness or physical exhaustion in the first instance, even if those may be a secondary factor.

As the poster in #4 points out, the participial adjective frazzled is almost the only surviving use of the verb. So that dated transitive Random House example wtte 'They frazzled me' is also an outlier, largely unidiomatic in the modern day.

The noun, similarly, survives almost exclusively in the fixed phrase worn to a frazzle, which LEO also seems to be missing. I would support adding it and will search out some context examples.

Meanwhile, yes, 'mit den Nerven am Ende' seems good.