DH and I found ourselves in the same position as you. Our landlord was selling the flat. We, however, didnt move out. We worked out that the flat was overpriced, and the fact that they decided to sell it with us still occupying it, we knew this would put a lot of people off. Our landlord was planning to give us the S21 once she had accepted an offer so that would have given us 2 months to find somewhere else.
We took the gamble and decided to stay and save hard in the meantime.
We stopped going out. I mean we literally stopped the work drinks, meet friends at the pub, etc...
We limited our takeaways to once every 6 weeks instead of every Friday.
We also played the downshift challenge where with our groceries, we bought a brand cheaper than we usually did and we saved the difference. We found some great products that we still buy up to now since then. Who knew that we would have preferred Tesco's own teabags to PG tips.
We saved all the £2 coins we were given as change.It was amazing how fast this builds up. 1 year later we had £750. A little annoying though when it came to banking the coins.
We also rounded up to the nearest £10 in our current accounts every evening. So if I had £56 left my account, I would transfer £6 to savings. I dont know if we could have been able to do this if we didnt have mobile banking. We hope to start doing this again but it will be transferred to overpaying the mortgage (we bought with a fixed high interest rate for our mortgage due to tiny deposit - interest rates have since fallen though).
We didnt buy any new clothes for 1 year.
I found that if transfer the savings money on payday, it is not there in your account tempting you. I obsessed over our household budgeting spreadsheet.
Packed lunches for work. Still do this now.
We eventually bought 1 year later; house was at the top of our budget but it was worth it purely for the fact that we knew we would only ever move on our own terms and not on the whim of our landlord. She was a lovely lady though and she responded to problems promptly. We enjoyed living in the flat but the constant worry of the S21 made us buy sooner than we'd hoped as we wanted to save a much bigger deposit. We are in SE if it means anything. I found that the rate at which house prices were going up was a lot faster than the rate at which we were saving.
Good luck with your savings journey and hope that your stay with the ILs goes ok. I know that I couldn't do it.