It's all relative as to what constitutes a lot.
We're aiming for 6 months of our household costs in savings, which is £15k. That's not pared back to the absolute essentials though. We're half way there at the moment.
To us, what we have saved isn't a "lot" as we have two professional wages (when I'm not on mat leave), but we also have a mortgage of £1k per month and two children.
It's the most we've had in savings since we brought our current house and renovated it, so I'm starting to relax knowing that we've got a safety net.
However my DHs job is precarious at the moment as the sector has taken a huge covid hit. He's so far survived two rounds of redundancies, but how long can that continue? So with considering that, it's not particularly "a lot".
I am aware that a lot of other people, could consider our current level of savings to be absolutely loads.
In your situation OP, I would say £7k after your purchase is fine, as long as your survey hasn't flagged up major areas requiring immediate work. You have some for moving costs, fees etc and if you keep saving after you move in, you'll soon save a bigger sum again.