If there is no mikvah building, but only a lake nearby, is it okay to use the lake during the day rather than the night (because of concerns for getting hurt or the water being too cold or the lake being closed) for removing Niddah status?
Immersing the day before the normal night is not possible, unless close enough to sunset (a rav should advise on details). So if by "during the day" you mean "the day after the normal night", then you are really asking three different questions
- Under what conditions can one delay mikva immersion?
- If one gets permission to delay, can a delayed immersion be performed by day, or must one wait until night?
- Can a lake be used as a mikva?
All three need rabbinic guidance. These are complicated issues of the utmost importance. I checked with a rosh kollel in Bnei Brak who specializes in nidda issues. Some sources which can help identify the issues are
Regarding delaying the mikva immersion, see Can a woman delay going to the Mikvah? on MiYodeya. Relevant issues are whether alternatives exist, whether the couple already has a boy and a girl and whether the husband agrees.
Immersing the day after the normal night is by no means obvious. There is an issue noted in poskim that a daughter will see the immersion by day and start to believe it is possible to immerse on the 7th day. Poskim will typically only agree if the issue delaying the immersion at night is valid every night for everyone (i.e., it is dangerous to immerse at night) but not when the issue is for a specific individual (e.g., there is a simcha one wants to attend one specific evening)
On using a lake as a mikvah, there are additional complications vs. a river. Specifically a lake might have somewhat dirty water which creates pressure to immerse quickly, might not provide enough isolation to be alone with the guardian and might not be deep enough to properly immerse. In addition one has to make sure there is a guardian, avoid interpositions (e.g., sand on feet) and wear a loose robe (see e.g., here).
When we asked similar questions, we were always strongly advised to find alternatives and NOT to put ourselves in such a situation.