At the time of the Crimean War, did the US have enough regular or professional troops ready for an invasion or could they recrute enough men fast enough?
They had 10,745 All Ranks in the US Army in 1854. Of these there were:
1574 in the Department of the East (from northern Michigan down to Louisiana)
1855 in the Department of the West (Minnesota to Arkansas)
2886 in the Department of Texas
1654 in the Department of New Mexico
1365 in the Department of the Pacific
And consisted of about 2,000 cavalry, 2,900 artillery and 4,000 infantry with the balance made up of military men not in fighting regiments.
Functionally speaking the British garrison of Canada dropped to two battalions plus the RCR at lowest ebb, which is roughly 2,600 fighting infantry (that's April 1855). However, as noted almost the entire US Army was hundreds or thousands of miles from Canada itself (and mostly a long way from railways) - and over the course of April-June 1855 the British shipped six infantry battalions to the Crimea (2/1st, 48th, 3rd, 31st, 72nd and 13th) and four cavalry regiments - if they all go to Canada instead then that's an extra ~5,000 fighting infantry and ~2,500 fighting cavalry, at which point the British have 10,100 fighting troops in Canada and the US has a total of 8,900 potential fighting regulars in the Americas!
This means functionally the US is going to have to rely on militia, and they don't have much luck raising good troops fast in OTL. TTL the Canadian militia (potential first-call out about 35,000 if the British can supply the rifles, and a few thousand permanent militia) will start training at the same time the US does, so by the time the US militia is ready for campaign they're facing about 50,000 defending troops and have lost their chance to take the positions at a rush.
You also have the perennial problem of taking Canada, which is as follows - the only really vital points for the British to defend are all on the far side of the St Lawrence (Kingston, Montreal, Quebec), and they have
GUNBOATS (quite a lot of them, actually - send most of the
Dapper class to fight in the St Lawrence and the US is never getting over that bloody river). Meanwhile the US is facing the problem of a blockade.
ED: it's not nearly as bad as the Pig War (where there were only a few companies that weren't
weeks of march from a railway line and the British Army was totally unoccupied) but it's not easy...
EDED: as an extra point of interest, the British were producing Swiss and German Legions for service in the Crimea. It would not take much divergence for them to be sent to Canada to train up if there's tensions or war, and they actually outnumbered the regular US Army - the German legion consisted of six light infantry battalions, three Jaeger battalions, and two regiments of Light Dragoons; the Swiss legion consisted of four light infantry battalions. The total strength was 12,978 officers and men.