Is my employer playing games?

oh dear you need advice ASAP.are you in a Union if so contact them. if not ACAS OR C.A. Agree that they SHOULD NOT have asked you to extend your sick leave!

If you extend your sick note, then you are playing into their hands. Claiming to be sick when you are not, could give them an excuse to fire you. Do not offer to extend your sick note. If the firm wants you to do this, ask them to put this wish in writing. Do not resign. If there is a Labour Office of sorts that can give you advice, contact them asap. You need assistance with formulating any communication with this firm.

Dependent upon your role and the time you've been there, there may be another option; I was in a similar situation position in respect of anxiety and through a union rep, was able to negotiate a golden handshake so that I could leave but with a few months salary to cushion me. It's called a compromise agreement, in exchange I got the good reference that I was entitled to, and money, and I agreed not to share my knowledge of the companies business.

If they were ready to have you working from home and the reason you couldn't was logistical I would suggest they should have allowed you to be back as available for work and get their act together to get the equipment to you.
If it was because they couldn't organise an OH appointment I would suggest this is something the HR /Wellbeing Team/Health & Safety department need to address as a matter of urgency. Can the OH consultation be conducted by phone perhaps?
Ensure you know how much sick leave you are entitled to and whether you may need time off in future, do you have holidays or unpaid leave you could utilise?
As others have mentioned I wouldn't resign, there are procedures.
Take care

Update: Firstly thank you all for taking the time to respond. I have spoken to my service manager. They have asked for an OH assessment and had asked for equipment to be sent to me but this has not happened for anyone not just me. It was suggested that I extend my sick note but I said I did not think I could do this as I had said I was fit for a phased return. I said that as from Monday I would be on no income. After some discussion it was agreed I go back to work, part time hours and they are happy to work with me and get my confidence back. I will be able to work from home until well into next year. I then had a call from OH and I have an assessment on my first day back. So it seems that all is well for now.
Once again thank you all.

When you loose your fur baby Chihuahua who was your companion and loyal friend of 13 years and you had to put him to sleep to stop any more suffering: The grieving is beyond belief:How do we cope to face each day without our four legged friend .im devastated heartbroken