you should be able to get her sen sooner by community physioptherapist for the lack fo weight bearing.
not for the wlaking as such because she isnt "delayed" yet for walking but clearly she should be able to weight bear.
my dd could weight bear from 12 months but still didnt walk til 25 months; my son didnt weight bear til 17 monhts, he also walked at 25 months - well first steps - he can now walk for miles.
both have hypermobility, hypotonia.
is your DD very flexible in her joints, all over or jsut in legs?
trauamtic birth does lend towards looking for CP - but tehr are also neruomuscular conditions which can afect trunk/legs more thn uppper body - you do need to consider trunk tone/strength as well, as this is important for walking.
you may be able to self refer to community physiotherapist for an asessemnt -ask GP to make that referral, because ultimately that is what she needs right now -some regualr physio exercises you can do with her while playing at home. dont wait for this push for a p[hysiotherpaist to see her - because whatever the underlying diagnosis, she needs some physio input.
so if you going to spend money yourself, get a private physio asssessement - look up childrens physio in your area (if you in london there are loads of good places, but there are others outside too).
the paed and diagnosis will come at some point anyway, but regardless of diagnosis you need to be getting her some basic physio right now, espec to look at her whole posture etc when stiiing as there is risk she gettting bad habits. it should be enough for GP to put "not weight bearing" as the "diagnosis" for now i n order tpo get the community phsyio referral for assessment -and this will also help the paed/neurologist in seeing where to go with diagnosis.
a physio can assess her physically and tell you where physically the issues are (tone, strength, tight ness or looseness, trunk, legs, etc) and more importantly can give you some exercises to do with her.
spend private money on seeing a paediatric physio for an assesment and advice, if you going to spend any - the diagnostics, tests etc - that can wait.
as riven said: having hand function and speech are the two most vital things .