In Europe (as defined above) it does not appear that any country allows an asylum claim based solely on the fact of being subject to obligatory military service. However, asylum claims can be made based on obligatory military service plus something surrounding the circumstances of military service. For example, during the Vietnam war, Sweden granted asylum to a number of Americans who either deserted from the military or who wanted to avoid being drafted, and asylum claims are granted for Eritrean refugees escaping the policy of indefinite conscription. The general policy as articulated by the UNHCR is that asylum claims should be considered when there is
persecution for objecting to military service for reasons of conscience, if applicable military acts violate international law, if conditions of service constitute inhumane treatment, if conscription is carried out by a non-state agency and the government provides no protection, or recruitment of children. None of these conditions exist in the Swedish or Norwegian military, so a Norwegian cannot avoid military service by applying to Sweden for asylum.
In Shepherd v. Germany, plaintiff, a US soldier serving in Iraq, deserted and applied to Germany for asylum, believing the war to be illegal. This is relevant because EU law allows asylum claims in cases of
prosecution or punishment for refusal to perform military service in a
conflict, where performing military service would include crimes or
acts falling under the exclusion clauses as set out in Article 12(2)
The court (CJEU) ruled that the plaintiff was not subject to persecution so the case was returned to the German courts, but in that ruling the CJEU provides extensive legal guidance on the connection between the "persecution" prong of an asylum claim, and military service under EU law.