They could do it tomorrow if they wished, no legal process required as it has already been done -
The Halton, Knowsley, Liverpool, St Helens, Sefton and Wirral Combined Authority Order 2014
The legal arrangements are such (in a convoluted way) that Halton Council had responsibilities for public transport which are transferred by virtue of the Combined Authority Order laid by Parliament to Merseytravel. So (in law) Halton collects public transport money via Council Tax and pays it over to Liverpool City Region. However, Liverpool City Region have then re-delegated responsibility for public transport back to Halton Council within the Halton area. So the exact same pot of money is then re-paid back to Halton. (I guess in reality it never moves anywhere, it's just a button on an accounting system). But Merseyside PTE (i.e. Merseytravel)
does legally include Halton within its area. - see articles 8 and 10 of the link above.
In 2015 and 2016 Halton did pay to re-paint all their bus shelters and stops in Merseyrail colours, and bus stop flags (but not bus shelter timetables, oddly) are rebranded as Merseytravel - see the attachments below.
However, they made the political choice not extend PTE ticketing arrangements into Halton. I recall reading when it was all set up this was going to be a transitional arrangement because the susbsidies payable are quite high and the "Merseytravel levy" on Halton Council would be quite high per capita as the population of the borough isn't that high (and is spread over a large area) but the number of tickets they'd sell would be quite low. But I did think the plan was to phase it in over time. Merseytravel's under 18's daily bus ticket "MyTicket" applies to Halton, for example.
When Bus Franchising kicks in, it's going to be even more anomalous if there are two different fare structures operated by the same transport agency, though.