My partner has left.

We each contributed 50/50 to household bills to the tune of £2000 per month.

My partner has also withdrawn £1000 from the joint account this month and has said they will continue to contribute £1000 (rather than £2000) until the house is sold.

Utilities are actually all in my name so they are technically not liable for anything other than the mortgage which is jointly held?

I'd like to remain in the house until next summer due to early exit penalties on the mortgage but I could attempt to move out and sell sooner (during winter) if I'd be unwise not to?

Do I apply for the council tax discount immediately or would this have some bearing on financial settlement?

No kids.

The council tax discount has no bearing on the final settlement so the sooner you apply the better. Your ex no longer has the benefit of living there but you do so some sort of adjustment to contributions may be reasonable enough. The current situation is unstable so there is an incentive to get it sorted.