I travelled on a group booking (15 pax) last year. We could not access the booking on MMB and could not OLCI. However we did check in via the self service machines in T5 on the day without issue.

With regards to seats, it depends if the booking was made directly with BA or through a Travel agent. Mine was through a TA and BA could not pre-assign seats for us which meant a few of us we stuck with middle seats etc. We also found that our seats were not all together but spread around the cabin in smaller blocks. The CSR at the time told me that if the booking is made directly with BA they have more control and will allow you to pre-select / purchase a seat depending on each person's BAEC status.

If you have booked through a TA they might be able to book specific seats for you, albeit at a cost no doubt. In my case the TA would not do that for us but YMMV as always.