The move represents a splitting of MHP's public affairs and corporate specialisms, although they will continue to work together on accounts when relevant.
Bloxham becomes head of MHP's corporate practice, a role vacated by Jonathan Lomax in the summer. He was one of the leaders of the new 60-strong Engage practice within Bell Pottinger, and had been with the agency for more than five years, having previously been a national newspaper journalist.
Lyons' co-head of public affairs Gurling has been with MHP since 2014, and has now been promoted.
Lyons himself was deputy MD of Bell Pottinger Political, and is also a former journalist. His experience includes six years as deputy political editor of the News of the World.
"The appointments are part of a drive by CEO Alex Bigg to better define MHP’s offering and strengthen its position in the market. They reflect his ambition for the agency, having recently hired Nick Barron from Edelman as MHP deputy CEO and Mike Robb from Cicero as head of financial services," said an MHP statement.
Bigg, who joined the firm just under a year ago, told PRWeek that splitting the public affairs and corporate offers will mean a "better identity" for both. "We are aggressive for both and we think we can get growth out of both."
Gurling said: "The prospect of Brexit poses British businesses with massive reputation and communications challenges. Andy and Jamie join us at a pivotal moment. Their experience in corporate and public affairs is a huge boost to MHPs capabilities and is great news for our clients. I look forward to working closely with them both."
Clients of the corporate and public affairs offers include L'Oreal, Chinese telecoms giant Huawei and train company Arriva.
On Friday, PRWeek reported that 23 of the roughly 175 former UK staff of Bell Pottinger had joined independent agency Pagefield, with most of these joining Pagefield's new sister firm. That same report also detailed hires by agencies including Celicourt, FTI, FleishmanHillard Fishburn and Lansons.
PRWeek also reports today on the arrival of three of their colleagues at Maitland.