1/ "9月底的预产期。因此邀请国内母亲首次来英探亲照顾。"
I hope you didn't disclose this information on your invitation letter, and didn't use it as your reason to invite your mum to England. Did you do that?
2/ "6月初在广州签证处递交的申请，遭拒（大概因为我申请的是一年多次往返签证）。"
In good practice, when your mum's visa was denified, the officer would give exact reason. What was it? "大概因为我申请的是一年多次往返签证" is only your guess. It shouldn't be used as an excuse not to award your mum's visa. If your mum is not sure of the reason, she can ring them to find it out.
3/ "再次申请获签的几率大吗？"
It depends. That is why I have asked you to figure out the reason of denying the visa. Next application should avoid the same issue, it doesn't necessarily mean to lie.
4/ "隔多长时间递交申请合适？"
No definite answer to this one. Your mum can reapply the next day.
Good luck.
