LCR are not looking to take on the 360. They have already released information on that (see relevant thread
https://www.railforums.co.uk/threads/liverpool-city-region-bus-franchising.282426/)
Yes the issue is that Liverpool City Region are planning to take a
much firmer line on cross-boundary services than Greater Manchester did, including cutting the high frequency and profitable commercial routes from Chester and Preston to Liverpool.
I don't see what would happen to the 360, or likewise to the 375/385 to Southport from Wigan. Once St Helens depot is franchised, the current operations will cease (since whoever takes it on will not be able to run it and won't be being paid to run it either).
Arriva could in theory apply for service permits to both Merseytravel and Greater Manchester and (if granted, which in the Liverpool region is not a foregone conclusion) continue to run those routes commercially, but from which depot, using which buses and which drivers? Warrington's Own Buses could take on the 360 (and would probably get a more positive reception being a municipal), but would they? And it's one thing for Rotala / Preston Bus to run Lancashire County Council tenders in the Ormskirk and Skelmersdale area, but another thing entirely to run an intensive service 22 miles end to end with a 20 minute frequency and approx 2 hour end to end run with no stabling facility en route so far out of their area.
The thing is this has been mentioned in consultation responses I have seen including CPT submissions to the Liverpool authorities, but they seemingly don't care on the basis "not our problem".