A word that means this time next week

From the Oxford English Dictionary

week, Phrases P2a(b)

(b) Following a specified day, as Monday week, this day week, tomorrow week, yesterday week, etc.: seven days before or after the day specified. Cf. earlier sennight n.

1680 E. Hookes Due Order Law & Justice 42 Mary Duncon, to have been called into the Court of Sessions this day week.

1889 ‘J. S. Winter’ Mrs. Bob i Let us say Thursday week, dear—This is Saturday, so it is quite enough notice to give. 1957 F. O'Connor Let. 19 May (1979) 220 Last Friday week I stood in a receiving line with your brother and sister-in-law for a good hour.

1990 Guardian 25 Sept. 15/1 Reilly..is due to announce his..squad tomorrow week.

2004 D. Peace GB 84 80 However, Monday week, there will also be a Union family rally in the town

Thus, the phrase the OP is looking for would be today week

As for sennight, according to quotes in the OED, sennight, one of its uses is the same as that of week, as described above.

My birthday is last Sunday sennight (made up)

But sennight is so British English and so obscure that it is best ignored except by language aficianados.