Congratulations I am in Leicestershire and cover all of the Midlands as a night nanny and have extensive twins experience.
Families book me for a date usually week before due date unless they are having a planned section then they know a date and plan for me to start 5 days post section as they plan to stay in hospital for that time. If I dont start on agreed date only 50% is charged and booking can be extended if I can fit them in (generally thats not an issue as I dont do back to back bookings as I like a break in the middle).
If you plan to breastfeed a night nanny will bring babies to you to feed and support and advise you on feeding and they will change the babies and re-settle so you can go back to sleep, once feeding is established the night nanny can give a bottle of expressed milk if you wished or you could continue to feed - entirely your choice.
One tip with twins have them in the same routine do everything together when one needs feeding wake the other and feed and dual feed them a night nanny can support and show you best ways to do this whether breast feeding or bottle feeding - for this reason with twins I would always use an experienced night nanny who has twins experience as its quite different to singletons. If you try to do one then do the other in the early days especially you will be feeding 24/7 (although I have strong memories of bf my little one for what seemed like 24/7)
There are many agencies you can use but most night nannies and maternity nannies have their own websites so its always worth doing a good google search and asking on here for recommendations.
When getting references right a list of what you want from a night nanny and ask questions related to that from the referees as will give you a picture of whether they will work for you.
You will know whether you can work with the night nanny or not from meeting them as you need to be comfortable with them and likewise them with you as they will be caring for your babies whilst you sleep and wandering around your home - I have been a night nanny for many many years and still always feels very strange on that first night with a family.
Dependant on what you can afford I have worked for a family who had 2 of us so I did 2 nights and the other night nanny did 5 nights and other families (which is more common) who have had 2-3 nights and then used a night nanny for a longer period of time as they rathered less nights a week over more time than cramming lots nights in and then having nothing (also had families start with 3 nights for 8wks then reduce to 2 nights for however many weeks and then sometimes down to 1 night to wean themselves off!!) - All babies I have night nannied for whether singles or twins or prem have slept through from dream feed (10.30pm) until 6/7am by 12-15wks.
sorry for long post - any questions please do ask.