Wow! There are so many POVs to consider here; it's not a straightforward situation.
First off, my eldest is 6 so I really have no idea what an 8 year old is or isn't capable of grasping. When I was 8 we didn't have a lot of cash about, and several of my friends didn't even get pocket money. It wasn't unusual if someone had a windfall that they would treat their friends. So I don't think that there was anything especially wrong in your son, letting his friend spend his money on him. As a child we would have judged the fairness of this on everyone getting an equal share, and if there was an uneven amount to be divvied up then the person who brought the bounty got a slightly bigger share. And that applied whether it was money, cake, sweets, or turn s on the skipping rope. If the amount here had been a fiver or a tenner, I don't think it would have been an issue. Would we have had enough sense as kids to realize that we shouldn't spend such a large sum? Honestly I'm not sure. We never had the opportunity to find out!
I'm also not really astonished at the amount they spent. A day out at a fair is an expensive day out. It creeps up on you. When we go we take a specific amount of cash and no more, and usually negotiate with the kids how to spend the last of it to teach them to make choices, and to understand that no more money means just that. (DH too as he loves to treat them and finds it tough to say no) Credit cards and debit cards have made money more abstract on a daily basis it can be very unclear to kids why you won't buy them what they ask for when you clearly have a magic card to pay for anything you want. When I was little we spent pennies at the shop, 1p = 1 sweet, and it wasn't hard to grasp. Now you have to start with numbers like 20 and 50 before kids have managed to count to 5, and why is a coin with a 1 on it worth more than one with a 10? It has become really complicated. I think I definitely had a better grasp of money at 3 than my 6 year old has now. I can totally see how a pair of 8 year olds, could spend that much money, and just imagine how much fun they had doing it.
OP, I'm wondering when you told your DS that he couldn't go to the fair, did you give a reason for that? I can imagine in my house, I would probably have explained that we couldn't afford to go to every fair…. that we had been to this fair recently….that we were going to that fair soon…..that it's expensive. And maybe one little boy told the other that his mum wouldn't let him go to the fair because she hadn't enough money, and then another little boy has a lightbulb moment and voila, problem solved, both little fellas head off the the fair, never imagining for a moment that anyone would have a problem with it. Pure speculation, of course, and I'm not suggesting that they be absolved from the responsibility but it's easier to teach children the lesson they need to learn if you can figure out what they were thinking when they did it.
As I see it, your DS shouldn't have gone to the fair without checking back in with you first to see if it was ok. If he decided to disobey you and go anyway, then deal with that. But it is possible that he figured that your reason for not going had been dealt with, and therefore it was ok to go. But he failed to consider that you wouldn't know where he was if there was an emergency, and that wasn't ok. That's not disobedience, as much as thoughtlessness. I'm not suggesting that there shouldn't be a consequence, but that it needs to be for the right offense!
As for the money, I think I would let him know that the other mum is asking that he pay back his share, and help him work out why this is such a big issue for the adults and what he can do to make amends (chores, etc.) Obviously, there's a considerable probability that I'm being totally naive because I really don't know much about 8 year olds and you are in a much better position to judge his level of awareness. But it is worth taking into account that children think differently to adults, and if you involve him in the figuring out a solution it will probably be a life lesson that he takes on board rather than a punishment that he resents because it seems unfair. Obviously if he understood what he was doing, and cynically manipulated his friend into spending all his money, then that's a different story. But in my experience most kids are a bit daft rather than devious. But maybe I'm about to get a rude awakening in the next two years!!
I'm a bit 
at the other mum demanding money off you, but in her shoes I'd probably be mortified that my child was out splashing cash around like that, and be apologizing for what happened. I have a tendency to blame myself for stuff first before I think it through properly. I suspect that maybe she might be the opposite and start by blaming someone else? But if she thinks that your son suggested/ requested the money (and I can easily imagine one 8 year old saying to another "did you get money for your birthday…why don't we use that?" being logical rather than underhand) and egged her son on to spend it then I can understand where she is coming from. I don't think it is totally reasonable to expect you to pay for something that wouldnt have happened if she had been more careful with the money/but neither is is entirely unreasonable to expect your son to make amends for his share of the crime. And if she is this confrontational with you, I can easily imagine her son being a little less than totally truthful about who said and did what and your son being painted as the villain of the piece.
The part I can't figure out is what she intends to do with your £45. How is her son going to pay back his share? Then does he get the whole £90 back, or does she keep it? Possibly it's one of those households where the kids birthday money gets absorbed into the housekeeping?
Reasonable or not, I'd probably pay her the £45. Maybe ask her over for tea by way of apology, let her know you were also shocked by what happened and have a chat about what you are both doing so that you can send the same message to the boys. Chances are she is frothing at what she imagines is your bad parenting, or your son's bad character. It might help bring it into perspective and it would be a pity if in the fallout from all of this was that the two boys couldn't be friends.