First off, realize this: they're all gonna laff at you!
Now that we've got that out of the way, here's some thoughts/facts on it:
-the Bobbin takes 590 wheels; 590s with the standard 38mm rubber are the same diameter 622 wheels with 23mm rubber, or within 4mm. So, yes, they'll fit inside the frame. However, the fenders will at the very least a l'il chunky, wrapped around skinnier tires. Theoretically, the fenders may rub the tires at their highest point, but tbh I'd be surprised if that were the case. More problematic would be brake reach; you'd need to change the calipers and, depending on what reach the stock brakes provide, and where the pads sit in stock form, it may be very hard (near impossible) to set up the brakes correctly.
-as far as hub spacing goes, the front will be no problem. The Birdie comes with a S-A 3 speed "freewheel" (ie, no breaks), which will be spaced between 116mm and 130mm; 120 seems most likely, but any of these are possible. So,you could run an aerospoke with the "track" rear hub, but you may need to spread the frame by 4mm (easily done) or put some spacers on the axle (also easy). Once you get the Birdie, measure the space between the rear dropouts to see what (if anything) you'd have to do. Obviously, if you go this route, your bike will become SS. If you wish to retain the 3speed gearing, you'll need to only change the front wheel.
Whatever you decide to do, be prepared to explain to ppl why you're putting a $800 track wheelset on a $600 40-pound townie bike. I'm not judging you; I ride a Worksman with a Chris King headset, so I am not fit to judge. I'm just trying to keep you prepared.