There is nowhere to stay at the springs- I saw one or two camper vans there, but there is no campground as such, and I saw no toilets or water. I thought that there was a camping about 10km west toward Yesa, but perhaps it is 5km as falcon269 writes. The springs are in the middle of a set of ruined buildings about 200m from the road-- the lower springs have got lots of hot mud with which you can plaster yourself. I think it is at marker Km336. There are almost no facilities along the north shore of the lake between Berdun and Yesa aside from a gasolinera (perhaps 8-10 km est of Tiermas) and the camping toward Yesa. There is a ruined village in the heights, but I do not know if there is anything there.
I think that the lake would be very cold in April, and likely not swimmable. But nohooligan can try and, if they live, can report back to us. The springs at Tiermas are delightful-- I think I was the only non=Spaniard there and people were very social and welcoming-- the young woman at the turismo in Javier told me that it is a secret (!) and so there is no official information on it. The more demure among pilgrims should be aware that the Spanish hot-springs aficionadi are sometimes informal in their attire, and that they must accept the local practice.