Sorry if I'm mistaken, have you already been on the holiday? I guess that might make things trickier?
We were in a similar position a few weeks ago, booked with loveholidays in March to fly in mid September. Late August got an email direct from the airline to say flights had been cancelled.
We had to be proactive and tell loveholidays our flight had been cancelled and what option we were taking (there was a section in manage my booking where you put whether the airline had offered alternative flights etc). My DH selected the option to say we wanted a refund but loveholidays came back to say we had to request this from the airline ourselves. I called the airline and they refunded the money to loveholidays.
I was really annoyed because loveholidays said we needed to speak to the flight team and be offered alternative flights (I'd already looked and knew there weren't any suitable so we just wanted a refund so we could book something else). They were vague about timescales even though it was less than 4 weeks til our holiday.
I got antsy after reading the reviews so I did moan on social media about the vagueness and the flight team got in touch the same day with an offer of unsuitable flights, I declined and requested a refund, which was in our bank within 5 working days. In all, it was a week from flights being cancelled to money back in our bank but we did have to fight to get them to acknowledge us in the first place.
We've rebooked with a different tour operator now.
It seems as though the communication with loveholidays is non-existent and I would be hesitant to book with them again even though our complaint was dealt with quickly (although I think that was only because I was so vocal).
I hope you get your money back soon, I'd definitely recommend using social channels to get to speak to an actual person as that seems to be the biggest challenge. Be polite but firm, tell them the solution you want repeatedly.
Good luck!