Why do nerve gas victims have chronic neurological issues?
Most of the nerve agents (and quite a number of commercially used pesticides) fall into the class of organophosphates (OP). Their mechanism of action is the inhibition of the acetylcholin esterase which leads to the build-up of acetylcholine in the body and thus a permanent activation of mucarinic and nicotinic acetylcholine receptors.
Irreversible inhibitors of AChE may lead to muscular paralysis, convulsions, bronchial constriction, and death by asphyxiation - and organophosphates can act as these. See here for some more information.

According to reference 1, four different toxic effects can be differentiated:  The cholinergic syndrome, the intermediate syndrome, organophosphate-induced delayed polyneuropathy (OPIDP) and chronic organophosphate-induced neuropsychiatric disorder (COPIND).

The cholinergic syndrome is happening in the acute phase of the OP poisoning and can be explained by the inhibition of the acetylcholin esterase. 

All quotes are from reference 1:


  In cases of human poisoning, general acute symptoms of peripheral
  nicotinic and muscarinic intoxication are clearly apparent. 
  These symptoms include miosis
  (unreactive to light); sweating, rhinorrhea, lacrimation, and
  salivation; abdominal cramps and other gastrointestinal symptoms;
  respiratory difficulties and cough; dyspnea, constriction sensation in
  the chest, wheezing; twitching of facial muscles and tongue, tremors,
  and fasciculations; bradycardia and ECG changes, pallor, and cyanosis;
  anorexia, nausea, vomiting, diarrhea, and involuntary urination and
  defecation. These signs and symptoms are accompanied by central
  effects such as dizziness, tremulousness, and confusion; ataxia;
  headache, fatigability, and paresthesia. Finally, seizures,
  convulsions, twitching, coma, and respiratory failure may also occur.
  If the subject survives the first day of poisoning, personality
  changes, mood swings, aggressive events and psychotic episodes
  including schizoid reactions, paranoid delusions, and exacerbations of
  preexisting psychiatric problems may also ensue.


The intermediate syndrome happens after the acute phase, around 7,7% of the patients develop it.


  The term Intermediate Syndrome (IMS) was first described by Senanayake
  and Karalliedde (1987) because it appeared in the interval between the
  end of the cholinergic crisis and the onset of OPIDP. Following
  exposure to various OP pesticides, clinical manifestations of IMS
  typically occur within 24–96 h, and affect patients without
  fasciculation or other cholinergic signs. The reported incidence of
  IMS ranges from 7.7% to as high as 84% (Shailesh et al., 1994).
  Although IMS is well recognized as a disorder of neuromuscular
  junctions, its exact etiology, incidence, and risk factors are not
  clearly understood. IMS generally occurs among patients with prolonged
  and severe inhibition of AChE, although not every patient with severe
  AChE inhibition develops IMS. Other risk factors of IMS include
  delayed metabolism of OP pesticides due to toxicokinetic factors or
  impaired organ function, severity of poisoning, elevated muscle
  enzymes, plus adequate or late oxime therapy. IMS has been linked with
  exposure to specific OP pesticides displaying the dimethyl phosphate
  moiety (e.g. fenthion, dimethoate, monocrotophos, dichlorvos,
  methylparathion) but has also developed after exposure to parathion
  (ethyl phosphate) and methamidophos (phosphoramidate; De Bleecker et
  al., 1993; Yang and Deng, 2007).


The organophosphate-induced delayed polyneuropathy (OPIDP) cones later and has been connected to the phosphorylation and subsequent ageing of the neuropathy target esterase by the OP (see reference 1 and 3). 


  OPIDP is relatively rare neurodegenerative disorder in humans that is
  characterized by loss of function and ataxia of distal parts of
  sensory and motor axons in peripheral nerves and ascending and
  descending tracts of spinal cord. The early neurological symptoms
  usually are sharp, cramp-like pains in the calves, tingling in the
  feet and hands followed by distal numbness and paresthesia. Pain and
  weakness in muscles spread rapidly and patients become unsteady and
  unable to keep their balance. Progressive leg weakness occurs,
  together with depression of tendon reflexes. Symptoms may also appear
  in the arms and forearms. Sensory loss may be mild. Muscle tonus of
  the limbs gradually increase and spasticity appears in the lower
  limbs. Physical examination reveals distal symmetrical and mainly
  motor polyneuropathy, with wasting and flaccid weakness of distal limb
  muscles, especially in the lower limbs. In severe OPIDP quadriplegia
  with foot and wrist drop were observed as well as mild pyramidal signs
  (Lotti, 1992). There may be some functional recovery in less severe
  cases with more distal involvement and sparing of spinal cord axons,
  but pyramidal and other signs of central neurological involvement may
  become more evident with time. The recovery affects only sensory
  nerves, while motor neurons may permanently lose function as indicated
  by Morgan (1982) who described the lack of improvement over 47 years
  in 11 patients poisoned with TOCP.


The chronic organophosphate-induced neuropsychiatric disorder (COPIND) happens to persons with a chronic exposure to OP and is most likely caused by permanent damages to the central nervous system (CNS), but the real cause is not known yet.


  The most common symptoms of COPIND include cognitive deficit
  (impairment in memory, concentration and learning, problems with
  attention, information processing, eye-hand coordination and reaction
  time), mood change (anxiety, depression, psychotic symptoms, emotional
  lability), chronic fatigue, autonomic dysfunction, peripheral
  neuropathy and extrapyramidal symptoms such as dystonia, resting
  tremor, bradikynesia, postural instability and rigidity of face
  muscles


References


Neurotoxic effects in patients poisoned with organophosphorus
pesticides
Long-term health effects of nerve agents and mustard
Neuropathy target esterase (NTE) and organophosphorus-induced
delayed polyneuropathy (OPIDP): recent advances