At first glance I'd assume that *Edward of Ulster continues his father's career in Ireland. Since his father made a bid for the scottish throne he may also do so (the early Bruce Kings were a little insecure on the throne).
Whether he gains the Scottish throne or not, he and any sons are likely to be declared Richard's heirs forestalling the Lancaster-York rivalry or possibly just postponing it a generation if the House of Ulster leaves no male heirs too.