Your 4 year old would go to pre-school (also called nursery school) yes but that is usually only mornings, say 9.30-12.30. Until September when they would start school (but that finishes at 3.30). And your 18 month old is a bit young for nursery school yet, it's more usual to start at age 2, 2.5 or 3 (you get 15 hours free nursery school per week from age 3).
Your options:
Full time nanny who does pre and post nursery for DC1 and all day for DC2. Expensive - 10 net per hour is standard london rate, that is probably 12-13 gross and then there is employer tax too. And you're paying nursery fees too.
Childminder - who looks after DC2 all day and will do pick ups and drop offs and pre and post nursery for DC1. Quite a bit cheaper than a nanny. However less flexible about hours and holidays and also you have to drop off in the morning, they won't come to you unlike a nanny.
You'd have no problem finding either I expect. Depends on your finances really!
If you think you'll go the childminder route, I suggest you first work out which nursery school and school your DC1 will go to, as most CMs will have certain nurseries and schools that they will and won't pick up from (because it fits with their other charges/their location etc).
Bear in mind that in some areas of London, nursery schools and schools are incredibly oversubscribed - you will need to do your research carefully to ensure you can find a place for your DC1. Sorry!