The territory that in OTL became the Louisiana-Purchase was given to the Kingdom of Spain in the 1763 Treaty of Paris, ending the Seven Years War. If there is no ARW or if it turns out to be a failure (probably due to a lack of french support) France will not gain back this territory. In this TL, there will also be no French Revolution in 1789, since there will not be OTLs coincidence of factors leading to it. Since France does not help the american revolutionaries, the substantial financial strains this meant for the French treasury will no be there and France will thus not be on the verge of bankruptcy in 1789, thus butterflying away the need to summon the Estates-General. Thus the representatives of the third Estate won't have the opportunity to proclaim themselves to be the National Assembly. And since in TTL there will have been no french troops fighting along the american revolutionaries, far fewer common Frenchmen will have come into touch with the contageous ideas of liberty, equality and civil rights. Those ideas will definately continue to be passionately discussed by proponents of the enlightenment in the cafés of the palais royal and the salons, but without the example of a successful bourgeois revolution in America, those discussion will stay mostly philosophical. A revolution still may come alomg later, but without coinciding with the crop failures of 1788 and 1789 and the following hunger revolts, it would likely run along the lines of the Glorious Revolution in England or the Revolution of 1830 in France with a fairly smooth transition of power.
Without the dislodging effects of the Revolutionary and Napoleonic Wars and also without the example of the successful United States, the iberoamerican colonies will stay much more docile and merely petition for some of the concessions and privileges the British granted their american colonies instead of fighting wars of independence. So by the mid 19th century, the americas would still be made up exclusively of european colonies. The question how much British North America will expand now depends on several factors. Since it can by and large only expand at the cost of the Spanish posessions, the question is: Will there be another war in the style of the Seven Years War? Most probably yes, since there have been wars like this every second or third decade in the 18th century. Will Spain be at war with Britain? Most Probably yes, maybe not in in the first one, but sooner or later Britain and Spain will find find themselves as members of opposing coalitions once again. Is Britain going to win this war? Most probably yes, since it was the most industrialised country in the world. Would Spain then loose territories in North America? Most Probably yes. The situation between Britain / British North America and Spain would be similar to the one between OTLs USA and Mexico. Taking into account Spains precarious financial situation, Spain might be inclined to sell parts of its possetions in North America even without losing a war, after losing a war it would be next to inevitable. As to Alaska, the Russians knew they couldn't hold on to it in OTL and therefore sold it to the USA. If they don't sell it in TTL as well, they will lose it one way or the other sooner or later anyway, there is no way for them to resupply it in a conflict with Britain and its Royal Navy.