Although you have right to unpaid time off work when a dependant dies that does not extend to a companion animal. Nowadays more employers appreciate the benefit to their employee by allowing them a day or two off work when their companion animal dies as most employees are not productive when grieving. Your employment contract may allow unpaid leave for a companion animal or it may be at your managers discretion, which is fine providing your manager is an animal lover, but they can refuse you time off. Of course not all employers have the luxury of allowing staff additional time off especially if there are 5 or less employees in the company. Where possible arrange time off even for a day or two.
If you do have to work because you cannot afford the time off or you have been refused time off, speak to your manager or a trusted colleague about how you feel and what information about your loss, if any, you would like them to share with your colleagues.
Of course some pet carers welcome a return to work as soon as possible to give them a distraction from the intense emotions.
If you feel you cannot go to work, you should make an appointment with your own Dr to discuss a period of leave from your work or a referral to pet bereavement counsellor.