I’ve had the same issue as you – with pretty much the same timeline (i’ve been proccesed form 08/04, started my application on 03/04 and travel 26/05)
I spoke to someone today, it cu tout after about 5 minutes but the woman was really nice and helpful – they have fast track option that you can use, but only from 2 weeks before you travel.
I used the number 0300 123 1973, I’d advise this number as they are helpful, just be warned that you will be on hold for around 1-2 hours.
If you stress to them that you’re concerned and, like me, you applied when the wait time was 3 weeks, they will put a note on your application so it gets processed quicker
I can’t say whether or not this is certain to work, but it does help to relax you a little bit
I’m going to ring back later on so will keep you updated as to how I get on
I think the number above will also help attach the reference number (i’ve not done this yet myself as I didn’t know I had to)