Hmm for one thing even if encircled I would expect the First Army pocket to leak a fair bit. Samsonov when encircled went into a deep depression and did little to break out. I don't see Rennenkampf being that resigned.
I would also point out that while not "destroyed" the Russian First Army was very badly hurt at First Masurian Lakes nonetheless so the delta is perhaps not as huge as you're postulating.
OK so on to consequences. I still see Ninth Army being formed on the left of the Austro-Hungarian First Army with Ludendorff's subsequent Polish counteroffensive. In OTL Stavka responded by assembling Fifth Army, First Army and a rebuilt Second Army to try to envelop the German Ninth Army. Now they have less resources to do that and may adopt a more defensive strategy.
Less well known the German Eighth Army tried to establish a bridgehead over the Niemen in late Sept but had their pontoon bridges destroyed by what was left of First Army. Likely they succeed in TTL Need to think more about how that plays out. There is the Russian Tenth Army to factor in.
There was also a failed German attempt to take the minor Russian fort in Osovets at this time (with 2 more failed attempts in 1915!) It can be argued that this siege may succeed in TTL.
Oh there is one other wild card to consider. There is no way the encirclement of First Army would happen without General von Francois' I Army Corps playing an extremely big role. It will be very hard for the Dynamic Duo to keep him from playing a major roll despite their personal dislike for him.