@EwaLilith
If you’ve not already done so I think maybe you should start your own thread so that you can get advice & support, because people are quite likely to miss you in amongst things here. If you’ve done that I’m sure that at this distance in time - & particularly in these circumstances - @Gil84 wouldn’t mind you posting a link here to a thread of your own just to try to ensure you catch as many people as possible.
Please don’t feel you’re failing your DD - you’ve done exactly the right thing by seeking a referral. 
Have you a partner/is your DD’s father involved in her care & upbringing? Sorry, that feels like a terribly rude question to ask, but if so, it’s important you get to the habit of asking yourself: “Am/would I blame Other Parent for This Thing? If not, why not?” Not that “blame” is a particularly helpful model really - but it’s really important, even if you’re primary caregiver, not to let yourself get tangled up in trying to claim [sole] responsibility for anything/everything less-than-good that happens to your DD. You just don’t have that kind of power; & you run the risk of actually failing your DD in some way if you’re focused on castigating yourself for things you’ve convinced yourself you’ve done wrong.
Have you contacted Scope &/or looked at their resources at all? You might find their forum useful & they may be able to advise if there’s anything you can do while you wait (some disability charities offer - limited - medical advice via helpline services). As far as the wait goes, if you’d be able to attend an appointment at short notice it’s perfectly reasonable to contact paeds & ask to be put on the list to be offered cancellations. If you’ve been given an appointment with a named consultant, contact their secretary (go through hospital switchboard if no direct number is given) rather than the general department & - again - ask about clinic spaces & if you can be offered appointments cancelled at the last minute. It might be that they don’t run a cancellation list, but if you don’t ask, you won’t know. If you’re asking to go on it, you do need to be able to drop everything to attend an appointment offered to you for the next day or sometimes even one on the same day.
You’re a third of a way through the wait for the appointment. Make sure you keep notes of any significant issues your DD has - & questions you have - between now & the appointment. If there’s anything you struggle to describe that’s unlikely to be reproduced in the appointment, it’s ok to film her to show the paediatrician. If they say they don’t need to see it, don’t worry/assume they’re dismissing you; being able to provide the option is what matters, but they may not need that additional information.
If you’re up to it, absolutely complain about the HV’s refusal to see you. When you say health visitor refused to ever see us, was that Full Stop, or “just” about these concerns? I know there have been issues with babies born during the pandemic not being seen by HVs as they’d normally be, but…
Don’t forget to look after yourself in all this, too: if you make yourself stressed & miserable with self-recrimination you’ll be much less use to your DD than if you’re sleeping & eating well & generally looking after yourself. If you’re struggling to do that, don’t be afraid to ask for help - that’s not failing [anyone] either.
I hope everything goes well with the appointment & the outcome thereof.