Don't be confused by the names.
Some p;aces call themselves 'Nursery', others 'Daycare' others 'Nursery School' others 'Playgroup' others 'Pre-school' but the terms are really interchangable.
The ones you have to stay are what used to be called 'Mums and Toddlers' but are now more often called 'Stay and Play' or all kinds of other names to be more inclusive - as of course Dads can go, and often Grandparenst, and sometimes childminders. They will often include babies too, not just toddlers. They are mainly for the adults to get out the house and talk to other adults, in truth.
You are looking for childcare. Your options are a Nursery (under any one of about 1/2 a dozen names), or a childminder, for most folk. On MN a Nanny is a 3rd optiion , but I think there is a massive London bias in that, as I've never ever met anyone who has employed a Nanny.
Community playgroups tend not to take before 2, but most private nurseries take from 6 weeks. Again, don't be confused by the language. 'Private' just means a business not maintained by the local authority - it is not the same as 'Private schools' - the vast majority of 'childcare' is 'private'.
To find out what is in your area, you can ask at your local children's Centre, or google 'Family Information Service' along with the name of your Local Authority. They will have a list of available childcare provision local to you. My advice would be to have a good look at 3 or 4 if you can - there are some brilliant Nurseries out there, but there are some I wouldn't leave a child in.