OP, if you're a single parent it's definitely not U to need some space from a three month old. However, a full day at nursery is quite a drastic step at that age, unless you really have no other option. Could you not get someone to come over to your house a few hours every morning instead? She could take the baby out in the pushchair or just occupy while you go out / sleep / whatever you need to do? Surely this would be cheaper and less stressful for you both?
It is true that your baby will come to no harm at nursery, but imo, it will be a highly stressful experience. At that age, babies are only conscious of being an extensionto you and possess no strategies to deal with separation. I'm a Child Psych and have studied the impact of separation at this age - most babies react by detachment to surroundings as a defence mechanism. It's not just the separation from you, all the noises of other voices, other babies screaming, different smells are highly stressful. I once read that it's comparable to an adult being left in a full football stadium all day. It's exhausting and a situation you survive, rather than thrive in.
If someone has to use full time nurseries at this age, at least the baby will have to adapt more quickly than if you put him / her in for the odd day. This will be totally disorientating at this age. Your child will be safe, but it is if no benefit to them.
I'm sorry if this sounds judgemental, but please read about attachment (John Bowlby is a start) and then decide if you still feel the same way.
I'm not saying that nursery will inevitably scar a child for life, but at 3 months, you could explore other options.