I'm in Berlin. My son is at primary school, and has ASD and sensory issues.
First off, you do know that are a few state-run primary schools which are bilingual for English and German, right? Are you aiming at admission one of those, or a purely German language school?
Bear in mind that the state-run bilingual schools go from 8 in the morning to 4 in the afternoon. Friday afternoons you can optionally leave a bit earlier. If your son needs Occupational Therapy, trying to fit it in around such a long school day can be hard.
German only schools are either Ganztagsschule (i.e., also from 8 to 4, although many have lessons only in the morning and afternoons are homework, childcare, or sport and hobby-type activities), or regular schools (i.e., 8 till about 12.30 or 1, and completion of a lot more homework expected than for the Ganztagsschule.
A very few schools start at 8.30 or 9.
From talking to parents who have dealt elsewhere with various issues of their child, and who have then come to Germany, it does seem very bleak in comparison.
As you no doubt expect, you will have to be very proactive. For example,
one of the boys in my son's class has sensory issues and ADHD. It hasn't been handled very well. For example, one of his teachers simply refused to accept that there was any issue, diagnosis, etc. And there the matter lay, with the teacher simply labeling the boy as naughty and disruptive, and not really bothering to teach him. That went on for two years.
GPs will probably be not much use. However, the German health care system allows you to directly make appointments with medical specialists. If the individual specialist says that they need a referral first, you should be able to go to your pediatrician and get the referral form (though you might need to insist quite a bit. Having official paperwork already with you from England should make this easier).
Obviously, ASD is not the same as SPD, but in the event that this is helpful, in my son's school, the process for him has gone something like this so far:
Diagnosis from an ASD specialist. Official report which I can use to try to get further help for him.
(The official report has also made it easy to get repeat referrals for Occupational Therapy).
Meeting with headmistress.
Meeting with a teacher whose job it is to look out for students who need special provision or assistance (providing special assistance is called foerdern in German, that teacher is referred to as the Sonderpaedagogin). NOT ALL SCHOOLS HAVE ONE.
Sonderpaedagogin fills out paperwork to apply for special status for the child. Child is observed in classroom by specialist teachers from an external school, their input was also needed for the paperwork.
Application for special status sent off. Wait for about a year (unfortunately, I am not exaggerating). Right now, we are in the waiting stage.
And finally, the bit that hopefully will truly be helpful to you.
Presuming that your husband's spoken German is better than yours, I would suggest that he phones the Schulamt in Berlin, talks to them about specifics of your son's condition and what processes should be followed.
By Schulamt, in this case, I mean various official centers whose job is to support inclusion of special needs in regular schools (the center's name is abbreviated to BUZ). One such center is in Neukoelln, and the link to it should be here.
www.berlin.de/ba-neukoelln/politik-und-verwaltung/aemter/schul-und-sportamt/schulamt/artikel.290419.php
There is another such center in Steglitz-Zehlendorf. If they have a web-page, they have certainly managed to hide it very thoroughly, but the phone number is (030) 902 992 511.