There is no difference between from now on and now with regards to formal or informal speech but there is a difference in connotation when using them.
When you say from now on you are giving a long indefinite period of time - maybe months or years to decades or even longer - and when you are saying now you don't indicate any particular time frame but you can only guarantee a very short time. It could be for a few minutes to maybe a few hours, or it could be a while longer.
With that in mind I would change your examples to
I fixed the tap in the bathroom; now it won't be dripping.
(You cannot necessarily guarantee an indefinite period of time with mechanical fixes)
I got a new job; from now on I'll have to wake up at 6
Our neighbours have left the house so now they won't bother you.
(The neigbours may only be out for an hour, or may be a little longer).