I have two DD. DD1 was born too early (35+2). No tongue tie but not strong enough to drink. Long story short: I could not keep up with supply and we had to top up. Felt so guilty. Is now a smart, happy and healthy 4 year old.
Now, when I look back, I don't feel guilty for combi-feeding DD1. We had a few weeks that we only used breast milk (as by this time she could breastfeed), but my production was not enough. So the top ups came back. I cried back then out of guilt...
DD2 is 4 months now. Had a complete tongue tie. Was nevertheless surprisingly getting enough breastmilk, but lots of problems with reflux and sleep etc linked to this. We were also referred. Things are better now, after the appointment. But somehow, magically, all this time I had enough supply. And why? I don't know.
So please, be kind on yourself! Try to sleep, eat and drink. Take care of your body after your emergency C-section. Pump regularly. This will all help your supply a bit. And accept that this may not be what you wanted, but it is the best that is possible under these circumstances! And that is enough.
And chase the referral, get that appointment. We had a certified lactation specialist at the appointment in the clinic and it was wonderful, all the advice that was given. So please, get help to get a good latch, especially after the tongue tie and lip band tie have been severed.