In OTL Margaret Beaufort exerted influence in the days after Henry VII's death - she was his chief executor and until her own death two months later was organising her grandson's coronation - if both are dead she will almost certainly move to ensure the succession fo her granddaughter Margaret.
James IV - is just like Philip of Spain want and expect to have rule over his wife - however I suspect the English will react in pretty much the same way - a series of restrictions on his ability to dominate the realm - in the 1550s they imposed those despite the desire of their sovereign to allow her husband a full role.
The key people are going to want a straight forward and immediate settlement - Margaret is their best option quite frankly - she is married to a fellow sovereign - she offers permanent peace with Scotland (any other choice means permanent war with Scotland).
The alternatives aren't going to attract a lot of support - Anne Lady Howard is childless, Catherine of York has issue but was in high favour with her sister's family and is likely to support Margaret (Henry VIII pardoned her husband shortly before his death and gave Catherine control of his lands on his death - she went on to become godmother to Princess Mary).
Margaret Pole (Clarence's daughter) - is not getting the Earldom of Salisbury because Henry VIII is dead - she was high in favour with Margaret Beaufort and is unlikely to try and claim the throne
Stafford isn't yet on the council and whilst he is wealthy he doesn't quite have the influence at this stage to mount a coup.
Mary Tudor had her own household with attendent staff but she is just 13 so - unless some noble grabs her (which is unlikely as I suspect she will be grafted to her grandmother's household on her father and brother's death) to try and mount a coup to take the throne in her name I think she's a non starter at this point.
Margaret is proclaimed Queen and begins a progress south - she is only going to be around two months pregnant at her father's death in April at most (Arthur Stuart was born in October).
Parliament is summoned in her name - they confirm her title and right to the throne etc - and attempt to impose restrictions on her husband - he is to be titled King of England etc - they ban him from appointing Scots to English royal posts etc - and so on - the succession guaranteed to Margaret's issue but in the event of her death without issue the throne to pass to her lawful heirs not to James etc.
And in this scenario Arthur born in October I suspect is named Henry in honour of Margaret's dead father and brother rather than Arthur perhaps.
What's interesting is that with Margaret as Queen of England her surviving son (if it's James V here as OTL) will be born Prince of Wales and Duke of Rothesay. He's likely to inherit both England and Scotland upon his parents deaths. Here James IV might live longer given that there won't be a war between the English and Scots. Margaret's not likely to have two surviving sons to split the crowns. So we'll see an earlier Stuart/Stewart succession.
We might also see English foreign policy change to reflect the Stuarts interests. Margaret's son will likely be betrothed to a French princess due to the Stuart's traditional ally/marital interest being France. James being heir to both England and Scotland will make him a much more attractive match. Though the French are a little short on royal princesses of an age with him given that several of Francis' daughters were sickly and died young. So they might not have much to offer. Could Madeleine de Valois survive the English climate? Or might we see a Spanish princess being offered as James' bride inlieu of Catherine of Aragon?
Though who will marry Mary at this time? Charles Brandon might not be close to Queen Margaret and thus might not have a chance to woo Mary. Mary's second (or third) in line to the English throne so it's entirely likely that we won't see her being offered to King Louis. Isabella of Portugal is a smart choice for Charles V to marry given that it'll endear him to the Spanish more. But Mary Tudor's extremely close in the line of succession to the English throne. It'll make that particular match a very tempting offer for the Emperor. Likewise, the English might not be keen on that match for the same reason.
Actually. Speaking of Catherine. What happens to Catherine of Aragon in this timeline? She's lost Arthur Tudor. Prince Henry is dead and no longer an option. Will the English return her dowry and send her home to Spain or will Catherine decide to stay as Dowager Princess of Wales and marry an English nobleman of royal blood? Or are we likely to see her being married off elsewhere by her father?
Arthur Stewart will be born in England, likely his ATL name will be Henry James and maybe he will survive because butterflies...
Margaret OTL had two other living sons after him, James and Alexander (with the last being born after James IV’s OTL death) and as James will not die like OTL but live much longer he and Margaret will have surely other children but James and Margaret’s eldest surviving son will be both the Duke of Rothesay and the Prince of Wales and will be King of both England and Scotland so we will have an earlier union of the crowns...
(Possible names for James and Margaret’s sons are James, Henry, Edward, Alexander, Robert, Arthur, David maybe also George, John or Richard for younger children while girls will likely called Margaret and Elizabeth).
Catherine of Aragon will be sent back in Spain and will be remarried at her father’s will (and seeing her as third wife of Louis XII is not impossible) while Mary Tudor will be married abroad (Louis XII of France is still possible as the future Charles V but for me the most likely match is Manuel I of Portugal as his third wife)
Here Eleanor of Austria-Spain will likely be married to her cousin John of Portugal and not to his father (who will marry either Mary Tudor of Catherine of Aragon if she had not already married) so Catherine will be free for marry Arthur/Henry Stewart.
The first option for James and Margaret heir will be Catherine of Austria, the second one a princess of Portugal and a French match will be only the third one on the list