I am not entirely anti medicine for depression but I have to say - what transformed my childs very serious MH issues (very severe anxiety leading to barely leaving the house, self harm etc) - was actually the basics of wellbeing
Her school put her in forest school every week, they helped her make new friends, I put a huge amount of time (stopped working) into getting her out socially and into woods, walks etc - community eventrs, helping her stay in touch with friends.
now - its much harder with a teen of course, my child was young enough to be taken around by me to a certain extent (but believe me - my child was seriously unwell and many weeks barely left her room) -
but - I do think that these things fixed her / helped her and healed her
all happened during the 2 years the NHS failed to give her even basic MH support
Depression isn't enough to get CAHMS help and if you get it - it will be shit anyway
I would focus on reading around improving MH/ depression - there are hundreds of books in yiur local library - and just implementing all the methods we know work - fresh air, community, connection , good eating, calm and rest.
Tell your teen it's okay there is no rush to be better - and just keep being there for them.
If you google low cost counselling you will find something near where you live - there are often centres in bigger towns that offer this.