Software is involved, but it isn't that simple.
The LA has to identify anyone who has applied for a school in another LA and forward details of their application to the appropriate LA. They also have to find out if anyone from other LAs has applied to schools in their LA.
The LA has to construct a list of applicants for each school and carry out address checks and check claims for sibling priority.
If the LA includes a category for pupils with special medical needs in their admission arrangements, they need to assess the claims for inclusion in that category. That involves getting suitably qualified individuals to meet and assess each application.
For schools that are their own admission authority (most of them these days), the LA has to forward the list of applications to the school so that they can sort them into order using their oversubscription criteria. As part of this, the school will need to assess claims for special medical need if they have such a category, faith schools will need to check claims for priority on faith grounds and so on.
Once the LA has all the sorted lists back, we come to the bit that can be automated. Software can work out who qualifies for a place at each school, see who has qualified for more than one school and eliminate all but their highest preference, and make sure that each pupil has no more than one offer. They may then allow the software to allocate places at the nearest school available for pupils that don't have a place anywhere.
The LA then has to notify any other LA that sent applications for places at their schools whether or not those applications were successful and wait to hear from those LAs whether the places are actually required - they won't be if the applicant got a higher preference elsewhere. Similarly, if any applicants named schools outside this LA, they have to wait to hear from those LAs if they applications were successful. If they were, they remove the local offer(s), which leads to pupils who missed out on the relevant schools moving up the list and getting offers.
When all movement has stopped and everything has been thoroughly checked, they are ready to make offers. Of course, even if they are ready a week or two before national offers day, they can't make offers then as the government has decided that all offers must be made on the same day.
The idea that you simply feed all the applications into a computer and it spits out the allocations in a few minutes is miles from the truth. There is a lot of manual processing involved.