Ephelant I wonder if you brother in law is worried by the depth of the commitment or perhaps not sure what the commitment is.
Traditionally, if the parents of a child did (sorry!) the God parent would become the child's guardian. My sister would assume that role for our kids if anything happened to us, so we made it clear to potential God parents that they would not be expected to step in if DH and I both died!
As Christians, we chose Christians to be the God parents, although they came from a variety of churches. It is my understanding that a God parent also needs to be baptised (in any church) and if not then a person who is not baptised can be called a 'sponsor', as a Godparent to a child, this is in the C of E. You may like to check I have got this right! For us it was not so much of an issue as my daughter and son were both dedicated and not baptised, but they each have God parents.
Anyway, my point is your brother-in-law, if not 'religious' or not a Christian may be unsure what he is letting himself in for. He may want to say no and be embarrassed to say no. Or he may be very busy and have simply put it off. I don't think you mention how old he is, and if he were relatively young I would understand this even more.
I would also say that as the uncle (only uncle) to your son he actually already has a very special role. The reality is his involvement with your son will be what makes the relationship and this may be better facilitated by shared experiences, like inviting him for a meal more often or going to sporting events etc together when your little one is older, or a shared interest in trains etc!
My in-laws do not live close and we have made regular commitments to meet up with them so they can see their niece and nephew and we try and do things that may interest them, but which have something for kids too. Not easy but worth it as the kids seem to really like seeing them.
This is maybe just me but a week is not long, not if you have a lot going on. Not long to make a big decision. Being a God parent is a lifelong commitment, not just a day. So it is reasonable for him to take time, especially if he was not expecting to be asked.
I think your husband needs to step in now, chat to his brother, make sure the brother knows it was a big decision to choose him (in that he should feel honoured in a good way, we thought you could do a good job) but it is not a problem to say no if your BIL doesn't feel it is right for him. I would also ask hubby to emphasise that at the end of the day what you both want is for BIL to have a good relationship with his nephew and you both and that whether he says yes or no to this 'responsibility' you just both want have a good relationship as a family with him, as wider family.
Lastly, I think traditionally (in the C of E at least) it is three God parents, two the same sex as the child and one the opposite sex - but you can have as many of as few God parents as you chose (within reason). And they can be male or female, so if you only have two chosen friends (rather than the traditional three) you could just go with that.
Good luck.