Plenty of misconceptions right here on this thread which answer the OP's question.
I'm a firm believer in reusable nappies/dialling down on the single-use plastics generally so am happy to use my experience to address the worries expressed on this thread.
Like many, I didn't know anyone who used reusable nappies before I had my son 9 months ago (and frankly even now still think I only know one person who does!), so it was worrying to make such a big financial commitment with no personal experiences to go on from people I know and trust.
We used a few disposables in the very early days, as these were sent to us for free by nappy companies, so this along with the above partly explains why people are anxious about reusables. People generally don't want to be the odd one out/the only one of their friends who uses reusables on their children (though we're not at school anymore, so I really don't get the peer pressure thing). Furthermore, nappy companies essentially indoctrinate people into buying their products before their baby is even born by sending them free samples. These companies generally have a much bigger budget for advertising and a much stronger presence in day-to-day life (e.g. a whole aisle devoted to them in supermarkets) so people are convinced that this is what they should do/what it is normal to do.
I understand that cost is an issue for many, but as several posters here have pointed out, there are so many budget options for reusable nappies (e.g. second-hand, using nappy libraries). The problem is that people just don't know about these options (though I believe they should be government-subsidised anyway). It could be that depending on the brand of disposables you compare with and/or the brand of reusables you go for, as well as whether your reusable nappies are new, you don't save masses on them for the first child – but if you have subsequent children you clearly win financially as you have already paid for all the nappies you will ever need. I will also sell mine when done with them so as to recoup some of the cost. For those who have asked how many you need, we have 24 and this is more than enough to go 3 days between washes.
In terms of leaks, all brands are different and all children are different shapes as well, which is why a nappy library can be useful. Practice also makes perfect a bit in terms of putting them on your child correctly so that leaks can be minimised. We use boosters at night (washable bamboo liners which go inside the nappy) which also help enormously and you can buy day versions as well. This sounds like 'loads of parts' as mentioned by some commenters on this thread but really isn't – in our case the nappies we use (BumGenius) are all-in-ones, in which we lay a paper liner to catch the poo and sometimes add a booster inside (e.g. for night-times). @WindFlower92 asked about different sizes and at least for the nappies that I use that isn't a concern – they have lots of different poppers on the front which you just open and close to adjust the size.
I'm lucky that my husband is supportive (both in terms of getting behind the reusable nappy decision and in terms of helping with some of the housework involving them – we both share the load of washing/lining them). If your partner is not supportive/you fear that you will end up doing all the work then chances are you would have ended up doing all the work anyway even if you used disposables. It's not the nappies' fault your husband is an idiot 
(ESPECIALLY if you have PND/mental health issues – your partner should DEFINITELY be helping you then.) Besides, 'all the work' amounts to probably an extra hour a week, tops. We wash our son's nappies 2-3 times a week and it's easy. Chuck them in the washing machine. Walk away. Put them in the tumble dryer or out on the washing line. Walk away. Then line them ready for the next go (takes about 5 minutes).
Environmentally, as others have pointed out you are not sending literally 1000s of disposables to landfill that will take hundreds of years to decompose. Even if you account for any environmental impact of washing and drying, you still do better out of reusables. Other more knowledgeable posters have already explained why in detail. For those asking if those using reusables on their children also use reusable sanitary products on themselves, for me the answer is certainly yes (Mooncup). I also use washable bamboo pads in place of cotton pads for using toner, removing makeup etc. In terms of travel, I take the train, walk or cycle wherever possible and practically never drive even though I have a licence. We have quite a few wooden toys for our son and many of his plastic toys were either second-hand in the first place and/or will be passed on when he is done with them.
Poo-wise we use paper liners to catch it so you don't have to go anywhere near the poo. Pooey liners get flushed down the toilet, but if the nappy is only wet we put the liner in the bathroom bin, just because the house we live in is old (built 1910) and I don't think the plumbing would cope if every liner was flushed. If there is still poo left on the nappy, you can either just chuck it in the nappy bucket and move on, or if you really want to, use reusable wipes or toilet paper to wipe off any excess, and/or sluice the nappy under the bathroom tap or in the toilet (while it flushes) so that when you chuck it in the nappy bucket it is a little bit cleaner.
In terms of smell, we actually find that disposables smell awful. Really chemical or even just like our son has pooed/weed when he hasn't. We haven't found this at all with reusables. My mother for one is extremely forthright and if our son smelt bad because of using reusables she would certainly be the first to let us know.
Another huge advantage of reusables is reduced nappy rash. My son has very sensitive skin (has eczema on his face and chest and has a very pale complexion) and compared to disposables he practically never gets it. I am also keen to be able to toilet train him earlier; I was reading recently (maybe on a Facebook group) about kids who are still in night-time pull-ups made by the big disposable nappy companies at the age of 6 or 7!! My view is that kids are just far too comfy in disposables and I would ideally like my son to be toilet trained by age 2 even if he isn't entirely dry at night (the 'readiness myth' is perpetuated by nappy companies as they want you to carry on buying their products; by comparison, 50% of the population, especially in Asia, is toilet trained by age 1).
I do generally tend to feel that we all need to do our bit regarding single-use plastics and that reducing/eliminating the use of disposable nappies is part of this. Even if you only have a few reusable nappies around you will be helping a lot. Reusable wipes are also a really good way forward. Again we used disposable ones initially, having received free ones from big companies (again!). When these ran out, we looked at our box of 15 reusable wipes and thought “this is never going to work, we're going to go through these at a rate of knots”. But you are genuinely more economical with their use – not just because there are fewer of them but also because they absorb more.
Someone mentioned that nobody wants to be handed a bag of dirty nappies when they pick their child up from creche. I personally don't mind at all – it's no different to emptying the nappy bucket at home. The creche my son attends has actually been quite supportive in terms of wanting to know how to put them on him etc even though he is the only baby out of 10 using them.
On that note, going back to work (which I did 3 months ago) hasn't affected my view of reusable nappies. Again, washing nappies takes time, but not my time per se. I can throw the nappies in to wash as soon as I come home from work (and again this is not every day – literally once every 2-3 days) and they're done in 1 hour while I get on with other things, such as playing with my son or doing his dinner. So I really don't see what difference it makes. All the old stories about the hours spent washing terry nappies are literally ancient history today.
I have also travelled with the reusable nappies. Vacuum packs are a lifesaver for fitting them in your suitcase! I don't travel with the full complement of nappies: I usually stay somewhere with a washing machine (e.g. my parents' place or an AirBNB) so typically just use the washing machine as a nappy bucket and chuck the nappies in directly. Might even wash more frequently (nappies plus any clothes my son gets through in 2 days, my underwear etc, all in the one wash) – so maybe every 2 days – meaning I travel with 10-12 nappies rather than the full 24 I possess. Once this is vac-packed down they don't take up that much space in a suitcase.
People tend to assume it's difficult without even trying – which seems a shame. And we don't exactly live in a huge house either (it's a 2-bed apartment with no outside space) so it's not as if we have loads of room to dry nappies etc either. If we can do it then anyone can.
I'm sorry this post has turned out to be so long but I remember some of the worries I had when starting to use reusable nappies and wipes, and hope this can encourage a few more people to give it a go. @gingerginger2's offer is incredibly generous and I hope someone on this thread takes her up on it.