On what grounds can I refuse to be trained in medication administration?
An employer does have the right to ask its employees to complete training, if the request is reasonable and within the employee’s remit (i.e., if as a support worker you would be expected to administer medication or if it forms part of your job description). In this instance, I would expect administering medication to fall into the remit of a support worker’s role.
If the request is unreasonable (i.e., the employer is asking the employee to complete the training in their own personal time) or you have a genuine concern about completing the training and administering medication, then you should speak with your employer to discuss these concerns. It may be that you can agree some adjustments to your role to avoid you becoming stressed and overwhelmed when administering medication e.g., buddying up with another colleague to do this for the first few months until you feel comfortable administering the medication on your own.
However, if you do not have any reasonable concerns and the training would fall into your remit as a support worker (which I believe it would), then the employer can ask you to complete the training and any failure to do so could be considered to be a failure to follow reasonable management instructions which may lead to disciplinary action being taken.