Set up makes a huge huge difference. So next week make sure he's had a decent breakfast so he's not hungry which will help. And make sure you have eaten breakfast. If you have a partner and can alternate so one does the early morning wake up and one does the swimming lesson then this is a good idea too. Try to minimise stress in that morning, so pack all the swimming stuff the night before and get him dressed when he wakes up, don't wait until it's time to go.
When you're getting changed, remind him of your expectations for going into the pool - I wouldn't force mine to hold hands in that scenario but if you want to, then include this. You might include things like "Remember, we walk slowly by the pool so we don't slip" and "We need to wait until the class before has finished" - basically you want your expectations to be in almost bullet point form so you can remember them all (even summarise with one word - hand - walk - wait).
If you think he doesn't actually know what is expected of him, then have a talk with him today, tomorrow, or at some point during the week, and maybe repeat this a few times e.g. watch a TV programme where they mention not running by the pool, read a book about a swimming pool and comment on not running, bring some small figures into the bath and "tell them off" for running, make them slip into the bath with how fast they are running etc to cement this idea of not running by the pool, so it really is just a reminder in the changing room. Don't try to teach a totally new concept in the changing room.
When you're walking in to the pool, keep these expectations in mind and any that you notice he's meeting, praise him "You're doing so well holding my hand, well done!" This will put him on a positive footing and remind him about the others, more so than a warning which often primes them to do the opposite.
Good luck!!