www.thestudentroom.co.uk/content.php?r=6825-Disabled-Students-Allowances-2015-16 should help explain DSA.
The idea of DSA is to cover costs that a student without disabilities wouldn't incur- For example, it might pay for a note taker for a student who struggles to take notes in lectures, or cover the cost of software that is especially useful for your condition. Things have changed recently, so it's now expected that students have to make some contributions towards equipment (for example paying the first £200 towards any laptop). However, it's definitely worth getting a DSA needs assessment done, as the assessors will have a lot more experience about what can help (and what can be provided) than you will.
For ASD/Aspergers specifically, it might pay for a mentor who could help with things like time management, and just talking through things they are finding difficult, or a dictaphone, if they might find concentrating in lectures difficult. However, needs assessments are based on the individual, rather than the diagnosis.
It does concern me in the OP's post that you were given initial poor advice by the learning support team. IMO, it's always best to declare a disability on the UCAS form. Unis won't discriminate, but they can take the diagnosis into account when looking at the application- and it can be confusing if a disability is mentioned in the reference which isn't declared elsewhere on the form. Do you think the learning support team seemed good otherwise? For some students, the quality of support at the uni can make or break their experience.
Offers definitely can't be withdrawn because you let the uni know about a disability. It's not like declaring grades, where not declaring a grade is considered fraudulent. The sooner you let the uni know about a disability/SEN/health condition, the sooner they can offer help and support to a prospective/new student.
In my experience, the absolute worst thing a student can do is not let anyone know about their disability until they are facing problems. There is a lot of support available at most unis for students with additional needs, but students do need to be proactive in seeking it out. Some students unfortunately leave it until it's too late and their problems have become very complex before they seek help.
It might be worth encouraging your children to post on the student room, and find out about the experiences of others with disabilities at uni (parents are welcome to post too).
It's worth bearing in mind that unis often won't discuss your adult children with you- they may have to do all/most of the communication with the unis themselves. If this could be especially difficult for them, it's a good idea to bring this up with the uni as soon as possible and see what they will accommodate. Again, if this is likely to be an issue, bring it up before a crisis occurs.