So things got better on the Sertraline but is now slowly getting worse again? Your dosage could be increased, it can go up to 200mg. If that doesn't work you may be one of the few unfortunate people (like me) who have to try multiple ad's before finding one that works, and for whom some ad's work for a while and then just stop. If this is your experience you definitely need to be treated by a psychiatrist and not a gp.
It's terrible that you can't access the therapy you need. I'm not in the UK so can't offer practical advice in this regard, sorry. Can your dh not fight on your behalf? That gateway worker must report to someone who can overturn her decision.
It's important to go back to basics in terms of self care when one is in the situation you're in.
Involved your husband and the dc's (if they're old enough) and plan a week's meals in advance. Try to make them as healthy as possible. Shop only for the ingredients you need to make those meals.
When you cook, try to make enough for 2 meals. Then either freeze or leave in the fridge for 2 days later. If the family doesn't like this, tough. Not having to cook every day is one less thing you have to stress about. Even better, let DH and the dc's cook.
Do DH and the dc's do their fair share of the housework?
Get some exercise. This doesn't have to take long or be strenuous - when DH comes home, leave the dc's to him and walk for 7.5 minutes. Then turn around and walk back. You've just got 15 minutes of exercise. If walking with the dc's is possible, take them with you.
Try to get out into the sun every day (weather permitting).
Try to find something every week that you can look forward to... 30 min coffee break with a friend, your favourite magazine, having a haircut... it doesn't have to be big things.
I haven't tried mindfulness myself but have read very good things about it.
Sit down and think about the things that make you snap at the kids. Write them down. Highlight the ones that are non-negotiable. Decide to let the rest go. This is very, very hard especially when you're very stressed and depressed as you just want them to LISTEN and DO AS THEY'RE TOLD and not expect you to think rationally about what you are expecting from them. Just try. Lower your standards. Try making lists for them so they can see what they have to do in the morning, afternoon, evening - use pictures or photo's for younger ones. Ask DH to help you set up situations in which you can do something enjoyable with your children.
This has become quite a thesis. I hope some of this is useful to you.