Assuming that the OS network stack applied the change immediately, then the packets will go through the current gateway.
The default route, and routing in general, is IP only and IP is stateless. It is just a packet mover. That is it gets a packet from IP A to IP B. There is no state stored between packets, that's for TCP.
TCP doesn't care what route it takes, the TCP state is based on the source IP/Port and destination IP/Port so your TCP connections should not be interrupted. That is unless your gateway is a NAT and changing gateways makes the IP packet go through another NAT.
The true test of this question would be to setup the set up the scenario and see what happens, but I assume you are asking out of curiosity.