In the UK Armed Forces, conscientious objection is grounds for a refusal at the admission stage and has been since the end of conscription in 1963. Where a person develops an objection to military service during their term of service, they have the option to appeal for a discharge.
Interestingly there's no primary legislation to manage this process but there are established military procedures to take care of this when it happens.
Mostly the process seems to be managed informally, with the objector simply being shuffled into a non-combatant role within their regiment and just left there for the foreseeable. Where that's insufficient for the objector, the process can be made formal, something that appears to invariably result in them being discharged from the service.
A member of the forces who has a conscientious objection is generally expected to raise the issue informally with his/her commanding officer. The officer’s options include rejecting the objection outright or moving the objector to a different position (such as a non-combatant role).
If the person concerned remains unsatisfied, he/she can make a formal application for discharge due to conscientious objection. After an interview which usually involves a chaplain or other third party, the commanding officer makes a recommendation to the chain of command where the decision for discharge or refusal is made. If turned down at this stage, the applicant can appeal to the Advisory Committee on Conscientious Objectors (ACCO) who hold a hearing and make a recommendation to the Defence Secretary.
Discharges due to conscientious objection are rare, with only six granted between 2001 and 2010.
ForcesWatch briefing: Conscientious Objection in the UK Armed Forces - Parliament UK
That all being said, a proportion of objectors are (apparently) removed from service without their conscientious objection being acknowledged. They're just ejected as unfit, in the same way that you would remove someone who was persistently ill.
The forces helpline At Ease reports that at least some who raise a conscientious objection have been discharged on other grounds such as for “service no longer required” or “unfit for further service”.