Somerfield are currently trying to sell off all of their loss-making Kwik Save stores in blocks. The only confirmed bidder seems to be Baugur (owners of Iceland) who have expressed an interest in some of the stores.
So some of the stores will trade as Iceland. The physical size of many of the Kwik Save stores means that many of them are two small to be of interest to firms like Walmart (owners of Asda) and Tesco. Morrison are unlikely to be seeking many additional stores as they are still rationalising their business after taking over Safeway. Sainsbury might want one or two of the stores but it's unlikely that they would be interested in many of them. Waitrose is one of the remaining 'big players' who are looking to expand their business but the customer profiles of Kwik Save and Waitrose are so different that Waitrose might find the locations of many of the stores unappealing (i.e. Waitrose want to put stores in areas where sun-dried tomatoes outsell baked beans whereas Kwik Save stores often serve the needs of less affluent customers).
If none of the companies mentioned above buy up blocks of Kwik Save stores, it's likely that the stores may temporarily close completely, pending sales of the premises one by one. At this stage, watch out for the two firms who are both keen to expand into smaller-sized stores, Aldi and Lidl.
So, in summary, the only company which is known to be moving into the old Kwik Save stores is Iceland but I'd be very surprised if Aldi and Lidl didn't take over quite a few.
Chris