I'm surprised your laundry company is using bambino mio wraps! IME they tend to use the pro wrap (it's quite a good wrap and bit bot more reliable and harder wearing than the mio wraps). That said yes, the pro wrap, rikki and mio wraps are all velcro fit. Is he taking his nappy off atm? If not you may find it is never a problem for you - my DD2 is a serial streaker and hardcore naturist but many people use velcro wraps right up to potty training without any problems.
I don't know what your nappy company charge you so I can't really say if a switch to DIY is the best bet but if you are getting leaks with their only offering then it is your best bet really. Prefolds have a tendancy to leak a lot more once your child becomes more active (but you can pin them to try to overcome this if you want to).
We don't use a tumble drier for our nappies - it certainly isn't a neccessity. Where are you located? If you had a local nappy project that operated a trial kit system then that would be great. How many nappies you need depends on how long they take to dry, how often you want to wash and how many you go through per day.
Microfibre is a quick drying option but quite bulky and prone to holding smells. Little Lamb do microfibre nappies - they are less bulky than fluffles (which are discontinued but still available in some places for now). Bamboo does take a while to dry but some bamboo nappies dry quicker than some cotton ones as you have to take the style of the nappy into account. Avoid hemp if you do not have a tumble drier.
It really isn't much hassle at all to DIY your own cloth nappies. I have two in cloth and I hardly notice having to wash them in amongst all the dirty clothes 3 young children and DH produce. I do 3 extra (nappy) washes a week at most atm with two in cloth but I wash a fair few of DS's nappies in with his clothes.
Would you be happy to buy second hand? Are you on a limited budget (bear in mind if you get good quality nappies they can be used for subsequent DCs), is a thin fitting nappy a priority.
The best advice is DON'T buy lots of one type but just try one or two of each that takes your fancy and see what you like best before buying in bulk. Buying second hand makes this a cheaper option as you can often resell for what you bought them for if they don't suit. If you want some advice on what might be best to try then let us know if you are on a limited budget (bear in mind if you get good quality nappies they can be used for subsequent DCs), and if a thin fitting nappy a priority. What do you like/dislike about prefolds? Do you have drying facilities outdoors? (who does in this weather????)
Sorry that is probably a bit long winded and rambling...