I am a regular substitute in elementary school and it is good that you are addressing this now vs later. The further up they go without the basic concepts, the harder school will be for the student.
I don't think it is so much to learn to be fast as it is to be efficient. We do give timed test for simple addition and subtraction about once a week to see how children are progressing. This is NOT a "test" and we explain it as a fun way to see how your are doing. Some children who don't do well will cry and it will ruin their day, others strive to be the first one done. Just last week, I stressed it was subtraction and 1 child came to be midway crying that she wanted to start over because she added instead. I explained to her.. the mistake was that she added the problems instead.. BUT... she got all of them correct.
We stress the UPS check and explain method on word problems. Showing work is part of the grade. If you can't show your work and explain, then we don't know how they got the answer.
A couple of math programs the children love are FastMath and SumDog. I don't know if they are available for the public.
Jumpstart has a series of games that our daughter loved growing up and it helped a lot with math.
Also, most schools have an enrichment program for about 4-5 weeks after school is out. You can take classes in a subject you need help in and you can also take fun classes like cooking, art, etc.
Our school offers a FREE afterschool tutoring for small groups which lasts about 45 minutes to an hour after school.
I would shy away from the big box tutoring companies because they are in the business to make money. Our district wants the children to learn the basics the way they are taught IN the school. Some children who go to the big box to learn faster do learn faster and then when they get to school they are confused because they can't do it the way it is taught which is the way they are graded.
Lots of practice, make it fun