Missing Kerias Shema, or Davening on time or with a Minyan due to Tevila:
Missing Zeman Kerias Shema: It is forbidden to miss saying Kerias Shema within Zeman Kerias Shema due to Tevilas Ezra [i.e. Baal Keri] and certainly it may not be delayed on behalf of the daily immersion. Thus, if one suspects that he may not be able to immerse with enough time to say the Shema on time, he is to say it beforehand. Likewise, if one plans to eat prior to immersion, he is to recite the Shema beforehand. In a case of doubt that perhaps one may miss saying Kerias Shema on time if he delays until he immerses, then by Tevilas Ezra [i.e. Baal Keri], he should say it with intent that it is under condition that if he does not manage to say it on time later on then he intends to be Yotzei with this reading, and if he does manage to say it on time later on after immersion, then he intends to not be Yotzei with this Kerias Shema. [Some Poskim, however, negate this ability of precondition. Practically, according to Admur one may rely to say it on condition.]
Missing Zeman Tefila: One is not to delay Davening within Zeman Tefila due to Tevilas Ezra [i.e. Baal Keri], and certainly may not do so for the sake of the daily immersion. [Nonetheless, those Chasssidim who invest much time in their Avoda of Tefila may delay Davening past Zeman Tefila until after they immerse for the sake of elevating the quality of their Davening. Nonetheless, they are not to delay Davening past midday even for this reason.]
Missing Minyan: Some Poskim rule that one is not to even miss Davening with a Minyan due to Tevilas Ezra [i.e. Baal Keri]. Other Poskim, however, argue and rule that one may skip Davening with a Minyan for the sake of Tevilas Ezra [i.e. Baal Keri]. Practically, one may be lenient in this matter regarding Tevilas Ezra. However, one may not be lenient in this matter for the sake of the daily immersion if he is not a Baal Keri.
If for whatever reason one could not immerse before Davening, is it of any value to immerse afterwards?
Yes. Some have a tradition that such an immersion helps retroactively.
 See Piskeiy Teshuvos 4:4; 88:3; Sheyikadesh Atzmo 63:11
 M”B 88:2; Divrei Yatziv 1:55; See Mishnas Yosef 5:16; Piskeiy Teshuvos ibid footnote 28; See Admur 613:19, Michaber 613:11, Rambam Yom Kippur 3:3 regarding a Baal Keri being stringent to immerse against Halacha that there is no custom to permit the forbidden but rather to prohibit the permitted
 See Admur 46:9
 See Admur Kuntrus Achron 70:2; Leket Hakemach 89:39; Sheyikadesh Atzmo 63:9
 Admur 46:9 in parentheses; Peri Chadash 46:9; Hagahos of Rav Akiva Eiger 46; Piskeiy Teshuvos 46:16 and footnote 170 in length; So rule regarding Sefira: Admur 489:12; Michaber 489:3 as explained in Taz 489:6; M”A 489:7 according to opinion that Mitzvos need Kavana; M”B 489:16; See Birchas Habayis 46:15 footnote 41; Piskeiy Teshuvos 489:13 footnote 62; Piskeiy Teshuvos 296:17; See Halif Lecha Shlomo 42; Salmas Chaim 64
The reason this Tnaiy helps even according to those who rule the Mitzvos do not need intent: Even according to the opinion who holds that that Mitzvos do not need intent to fulfill ones obligation, if one specifically has in mind to not fulfill his obligation through this counting then everyone agrees he is not Yotzei against his will. Therefore, if one remembers at night he is to count with a blessing. [Admur 489:12; Rama 489:3; M”A 489:8; Chok Yaakov 489:14 Ran Rosh Hashanah 3 in name of Rabbeinu Shmuel; Beis Yosef 589; Erech Hashulchan 489:5; See Kaf Hachaim 475:59] This is unlike the opinion of the Reah who rules one is Yotzei even if he screams that he is intending to not be Yotzei. Practically, we do not suspect for his opinion. [Beis Yosef ibid]
 Daas Torah 46 that the ability to condition only applies by Sefira which is Rabbinical; Taz 489:6 regarding negation even by Sefira; It is thus questionable whether he has fulfilled his obligation even if he made a Tnaiy and remembered later on. [P”M 489 M”Z 6]; See Piskeiy Teshuvos ibid
 Although Admur ibid writes his ruling in parentheses, which seems to indicate doubt [perhaps due to the arguments presented by the Daas Torah], practically, one may do so as is the simple implication of Admur 489:12 regarding Sefira where no differentiation is made between Biblical and Rabbinical
 M”B 88:2; Kinyan Torah 7:4; Piskeiy Teshuvos ibid; See Admur 613:19, Michaber 613:11, Rambam Yom Kippur 3:3 regarding a Baal Keri being stringent to immerse against Halacha that there is no custom to permit the forbidden but rather to prohibit the permitted
 M”B ibid that it is possible that one should be stringent even regarding Tefila Betzibur
 Mishmeres Shalom 2:1; Halacha Lemoshe Tevilas Ezra 36; Meorer Yisheinim 88:16; Minchas Yitzchak 4:61; Kinyan Torah 7:4; Teshuvos Vehanhagos 1:109; See Divrei Yatziv 1:55; Beir Sarim 4:126
 Piskeiy Teshuvos ibid
 Kinyan Torah 7:4; Piskeiy Teshuvos ibid
 Kinyan Torah 7:4 in name of Pischa Zuta; Piskeiy Teshuvos ibid footnote 65