c4 is generally not a good idea when you king is not safe, but I don't think that pawn push alone is what caused your collapse. It happened because you were not making threats. You didn't play g5 when the only thing he can do was take back because his knight would be under attack. Instead of doing this active move, you moved your bishop, which, though having the merit of threatening to trade of your light squared bishop for his, thus getting rid of the piece you somehow thought was doing nothing for a bishop that you evidently thought was doing a lot more than yours. In hindsight, remember that this light squared bishop of yours is a solid defensive piece, that could be used later at the opportune time, after you did something, like play g5. Since you played this move, your opponent had time to move his knight, but he didn't move his knight at first, first he played, on move 12, played a6, which should have caused alarm bells to be ringing in your head. I think, because you failed to play a4, in response to a6, is the real reason why you lost this game, rather than playing c4. Although c4 didn't help your situation, it did however, give your opponent time, to come up with a plan, to undermine your center with flank pawns, which he did, and created a strong passer and won because of it. 16... Nd5... This move shows that your opponent knew exactly what he was doing. This is a move that causes you to want to move your powerful bishop so that it won't be taken to allow you to get doubled pawns. However, maybe you could have saved the day by ignoring this move and playing a4 nonetheless, and allow him to double your pawns, and trade of your powerful bishop. But, then again this will allow him to put rooks on the b file which was unavailable to him earlier due to the dark squared bishop. You had bad options, allowing for doubled pawns to pay a4 may or may not have saved you, but, you see why I say that one must play active moves. c4, your move, doesn't make any threats. His move, 16... d5, made the threat of taking the bishop and thus doubling your pawns, which forced you to spend the time to move your bishop out of the way, which gave your opponent time to play b5, which allowed for him to create strong passer which won him the game. You played g4 in hopes for a kingside attack, and later did not go through with it, and wasted time in the center, which gave your opponent time to launch a decisive blow with his flank pawns, and your kings safety played a role to because it was under attack, because you chose to keep it in the center after deciding to open up the c file, which are two ideas that don't go together. Yeah, all of this was pretty much said already I just elaborated a little and adding some other things to keep in mind. I'm not much better at chess than you but I noticed I can analyze games very well, and I hope you believe me when I tell you that your kingside attack was without merit. I played through it and found that your opponent had a lot of counterplay, despite the fact that you could temporarily create a situation where two pieces are attacking his king, however, the trick would be to get rid of his knight and make black forget about his g pawn and not push it? If you could somehow control his mind? 
I play someone who uses the London system and I notice whenever I develop my light squared bishop to f5 and it gets traded off he usually doesn't have enough steam for any attack. I'm not saying bf5 is the solution for the London system, I am just saying it's hard to get enough pieces to the kingside if your light squared bishops both get traded off. I'm sure there is an attack I do not know about that combats this, but in this game, it looks like white had nothing on the kingside attack, and black can go on to win with an attack on the queenside that would slowly be commenced throughout the course of the remainder of the game.
My guess is that when your opponent plays Bf5, g4-g5 is not the answer. I could be wrong, and I'd be happy to be proven wrong because I'd like to be corrected because I myself am a low rated player that wants to learn. So if anyone has any games where white won by playing g4-g5, even after both light squared bishops where traded off, please show me, and I will look into it, and stand corrected if need be.
In this situation though, the g4-g5 idea didn't work, though it could have worked if your opponent blundered, which is entirely possible in a low rated game, so you ought to have gone for it. Maybe you were waiting for your king to be safer, maybe you thought by playing c4 magically things would have gone in your favor because you know GMs played it before and heck if they did it it ought to work for you.
Whatever the reason was, you delayed g5 long enough, to allow it to slip away, and you neglected to stop your opponents plan after he played a6. The plan after a6, was to play b5, like your plan after g4, was to play g5. I'm not trying to be mean, people miss this stuff all the time. I myself miss stuff. We miss stuff because we fail to see that it is a critical moment, when your opponent played a6 it was a critical moment, you needed to act with a4, but you didn't and played something else. Turns out everyone at our rating level has the problem. We fail to look for counterplay after we see that we can take a pawn. We fail to see the relavance of our opponents last move because we are too busy with our own ideas, etc. I myself am trying to figure out how to stop making these types of mistakes, or at least lower their frequency. I know the key element of getting better is to play games with longer time controls. I don't know what time controls you where playing in this game but if you are not already playing long games you should consider starting, because, you can't get better if you do not play them. I learned this the hard way. If you are not playing long games, do yourself a favor and play them, because coming from a guy who plays mostly blitz, I have been playing for over 15 years, and I am still below 1500 rating. Play long games.
Everyone has this problem, don't be discouraged. You should look at GM games with the London system where Bf5 was played by black, and see what they do. That is my advice if you still want to play the London system.
I thought some good points were brought up by IMBacon. They're no longer visible to me. Good thread (IMHO) because many players find themselves in similiar situations very often. Asking questions is the smart thing to do. Valuable input was lost here with some answers removed.
I am glad others find this thread useful. I can't see IMBacons posts too, don't know why they disappeared...
He often deletes his posts... sometime it appears to be pretty random to me, but yeah, he does it pretty often and I don't know exactly why. Sometimes it's because he gets into minor verbal jabs then I guess later regrets it, but I don't know why he deletes something where he gives advice.