Hi OP, I am a swim teacher and you've had some good suggestions on here.
I would just add a couple of thoughts.
Firstly, at her age persistence is key. I know it's not always possible to go once a week, but regular visits will definitely help.
Also, given her age, it's quite normal to be clingy I promise.
I would go into an adult pool if you can because that way you can hold her with less problem than in the shallow children's pools.
Start by letting her sit on the side until she's happy and comfortable. Then bring her into the water with you.
For the first couple of times gently bounce her around the pool. You can use toys and sing songs to make her feel more comfortable.
Gradually move her around more vigorously as she gets more confident and try trickling some water on her face so she can get used to it. Always tell her so she can ready herself - try to limit surprise splashes as far as possible.
Once she is happy like that you can start holding her further away from you or to the side of your body to increase her independence.
Welcome to PM me if you have any questions 