You cannot be furloughed for health reasons presumably because you are not extremely vulnerable. That is just the law, and your employer is following the law. Here it is btw:
"You can be furloughed if you’re:
- on sick leave or self-isolating because of coronavirus, speak to your employer about whether you’re eligible to be furloughed – you should get Statutory Sick Pay (SSP) as a minimum while you are on sick leave or self-isolating where you meet the qualifying conditions, if you’re furloughed while on sick leave or self-isolating, you will no longer get SSP but should be treated as any other furloughed employee
- clinically extremely vulnerable (at the highest risk of severe illness from coronavirus) and are unable to work from home following public health guidance
- unable to work (including from home) due to caring responsibilities because of coronavirus, such as caring for children at home because of closed schools and childcare facilitates or caring for a vulnerable person in the household"
So it seems you are not eligible to be furloughed for health reasons. Your employer CANNOT furlough you unless you meet one of the above conditions or if there is no actual work for you because of the pandemic.
The second SEPARATE issue is your claim/right to work without a mask. Your employer has the right to insist that all employees wear a mask under certain working conditions - healthcare would be one of them. You can ask for a copy of their policy on this. It is worth noting that they cannot discriminate against you because of your health conditions UNLESS they also cannot allow anyone for any reason to not wear a mask for health and safety reasons.
This claim seems to all hinge on this: I have health issues which mean I cannot wear a mask for extended periods,.
Your doctor apparently does not agree with this with enough certainty that he can write you an exemption note on this basis. I would say that the important word here is 'cannot' rather than 'would find it difficult or uncomfortable'. What he has done is provide a letter that lists your health conditions so that your employer can check for themselves that they are not discriminating against you as noted above.
(If you have been specifically told by a doctor that you cannot wear a mask for extended periods of time then you need to go back to THAT doctor and see if they will write you an exemption note.)