In the grand scale of things, whether we will have to pay roaming charges for using mobile phones in the EU after Brexit is a fairly minor issue.  Sometimes, though, it is the apparently minor issues which can highlight the scale of a problem.
The EU legislation which abolished roaming charges is based on capping the amount which mobile operator A in one state can charge to the customers of mobile operator B from another state, but it does not apply to states which are not party to the agreement, so that operator A is free to charge a great deal more, if it wishes, to customers from third party states.  If the UK fails to strike a deal which keeps the UK in the Digital Single Market, (and the UK Government has made it clear to date that it wants to leave the single market) then it becomes a third party state for the purposes of that agreement, leaving the EU operators free to reintroduce higher charges for UK operators whose customers travel within the EU.
Will they do so?  At first sight, there’s no necessary reason why they would choose to do so – if the choice is entirely up to them.  Life isn’t as simple as that, however.  There is potentially a problem for EU-based mobile operators if they want to treat one third party in a different way to other third parties; it could be seen as an unfair trading practice leaving them open to legal action.  In the absence of even a more limited agreement on the specific issue, that fear may lead them to seek to charge UK operators on the same basis as other non-EEA members, which would mean an increase in costs for UK mobile operators when their customers travel to the EU.  Conversely, of course, the UK operators could charge EU operators in respect of their customers travelling to the UK; any net increase in cost therefore depends on the net difference in travel between the UK and the EU.  That difference is likely to work in favour of the EU operators.
UK mobile operators can choose not to add roaming charges to our bills, of course (the big operators have already said that they won’t); and the UK Government can in any case act, as it has promised to do, to legislate to prevent them doing so or limit the extent to which they can do so.  But UK legislation can only control how any charges levied by EU operators are passed on to end users, not whether they are levied at all.  If UK operators face increased charges, they will undoubtedly pass them on to users one way or another; if not through roaming charges, then through a more general increase in bills.  Any government ‘promise’ that roaming charges will not be introduced is only part of the story; they are simply unable to make any promise that the costs of using mobile phones will not increase, one way or another, as a result of Brexit.
As I said at the outset, this is apparently a very minor issue, but the complexity serves to underline how little real work has been done on the detailed implications of the decision to opt out of the single market, and government attempts to pooh-pooh any suggestions that there will be problems demonstrate an alarming level of either complacency or a refusal to face facts.  It is the UK which has decided to leave the single market; continuing to demand that the EU treats us as though we are a member whilst not being bound by any of the obligations is leading us into a wholly unnecessary state of chaos.  And this one small issue is multiplied many times over in a whole range of other fields.