In my experience agencies use an extrnal freelance social worker to do their form f assessments so that workload does not affect things in this way. My start to finish was 4 months so this is certainly possible, 6 months more usual. I would read them the riot act and give them one last chnace to get new appointments organised which have to be kept to or you are off elsewhere.
I often see comments that LA's only place difficult children with agencies etc. This is completely untrue. LAs will always make placements with their in-house foster carers first if at all possible due to the HUGE cost savings. However invariably they do not have enough in house people to cope with the population of children who need homes. This is particularly the case with larger sibling groups.
IThe two sibling groups I have had (from two different LAs) were completely "vanilla" ie no medical or behavioural problems at all. The srvice and support we receive from our agency is excellent and the case load of our agency social worker is a fraction of that of the caseload of a LA social worker so they have more time for you.
LA foster carers can be sure of having 100% occupancy but in return receive far less remuneration.
If you can take larger sibling groups you can be fairly certain of being full-up most of the time anyway and the greater remuneration means that even if you have some void periods your annual income will probably still be greater
Financial considerations can be something of a taboo subject when talking about fostering but given the choice of providing a good home and care for less money or more i will take the extra money.
I have just discovered that the agency I am with is a relatively poor payer and I can take home and extra £650 - £1000 per month with other agencies without taking on more "complex" children. Of course I am going to explore that.
The worry for you is that if your agency are so inefficient on the admin side that it may cause LA's to be reluctant to place children with them. Speak to someone senior at the Agency and be open and honest with them.
Good luck. There are lots of sibling groups waiting for homes. It is really heartbreaking when the case notes read "home sought for three brothers or if not possible (eg) the 7 year old can be separated from his brothers".