The first lion rescue, which took place in 2010, brought 13 African lions to the park.
Its success encouraged the park to take on a second mission in 2024.
Aysa and her cubs, Emi, Santa and Teddi, now nearly three years old, travelled more than 2,000 miles across six countries from Poland after being abandoned in the Donetsk region of Ukraine.
Ms Williams said: “We are extremely proud of our previous rescues, and it is rewarding to see all of the lions settle into their home in Yorkshire.”
Park CEO John Minion said the support from donations had been “incredible”.
He added: “We are looking forward to welcoming the new lions to the park and proud to be rescuing lions for the third time.”