In his paper Constructions in algebra (MR349648), Seidenberg fixes some errors in Grete Hermann's classic paper Die Frage der endlich vielen Schritte in der Theorie der Polynomideale (MR1512302). Most of the work concentrates on two problems:
- Effectively compute the primary decomposition of an ideal.
- Effectively compute the associated prime of a primary ideal.
Together, these do give a primality test: first check that the ideal is primary by computing its primary decomposition and then check that it equals its associated prime.
Seidenberg's paper is very well written, but he obviously has a very limited audience in mind. He does prove along the way the existence of primitive recursive degree bounds, but the task to put them together into a precise form would require a fair amount of time. Seidenberg's constructivist point of view gets in the way of pragmatism from time to time.
There has been a lot of water under the bridge since 1974 and much of the contents of Seidenberg's paper must have been redone for practical use in modern computer algebra systems. Since working out the bounds from Seidenberg's paper looks impractical, I would consider looking at the computer algebra literature to see which algorithms are currently used to carry out the two steps above and derive effective degree bounds from these.
I am not very knowledgeable in such practical matters, but there are probably computer algebra experts lurking around here. If they don't see this post, then I suggest you ask another question for algorithms or degree bounds for steps 1 and 2 specifically.