Hiya,
I changed to washables when DD was sbout 6 months old, and haven't looked back. I was in a flat without a tumble dryer too.
How many disposable nappies do you go through on an average day? My DD was on about 4-5 a day (a change about every 4 hours, and a couple spare for poos after just changing ). I had 12 daytime tots bots which gave me enough for 2 days, wash them and the the third day while the others are still drying you have enough left.
I only had 2 night nappies, and ended up in disposables for one night out of 3, and I think ideally at that age we could have done with 4 night nappies.
Wraps - I had 4... She would have the same one for a day and night (unless it got leaked on), and I would wipe the inside out to keep freshening it. Some people use fleece or wool for night when they don't during the day, so that would be different. You need enough to have at least one spare so that when you have the violent poo you can change the wrap too!
That is the very minimum needed - if I had more pennies I would have had a few more wraps and nappies.
Washing machine on constantly? Nope. Without tumble dryer it was on every 2 days with nappies, with tumble dryer - every 3. So an extra 2-3 washloads a week. By the time you're not having to wash so many clothes because of disposables leaking, its not much more at all!
Nappy bin... have you seen/got the nappy wrapper bin? I just took the plastic bag dispenser out of mine, and used the bin as a bucket - it is ideal because the lid locks in place stopping investigating kiddies! Its fairly narrow so doesn't take up too much room.
Liners - disposable liners are available from boots, tesco etc. Reusable ones are all over the internet.
I didn't soak my nappies - I rinsed any poo off, and then left them dry in the bucket (a few drops of tea tree oil & lavendar on a cloth in the bucket makes it smell better). You do have slightly more staining like that, but I seemed to have stains when I soaked too.
Wraps - I put them in with the nappy wash, where you don't have as many cos you only use one or so a day, it all balances out.
I would highly recommend going to www.thenappylady.com and filling in her tailored advice questionnaire - you answer a pile of questions about your lifestyle, and what you want from the nappies, and she advises the best type for you, and how many you would probably need.... Remember though - it is FREE advice, you do not have to use her shop, and there are often better deals around the internet.
Good luck!!!