I can't read the whole article as I don't have Telegraph premium but I do agree that it'd be a wise move to increase defence spending.
American will almost certainly still be our most important ally even with Trump as POTUS but there will always be occasions where the interests of the UK don't line up with the US (as demonstrated by both the Suez & Falklands conflicts in the past).
Currently it'd be very difficult, if not impossible, to run a large operation without US support; most NATO nations under spend on defence & only France has an real capability & will to use its armed forces. The German's, since reunification, have fallen away & no longer have much will to commit to defence. I'd like to think they'd respond militarily to a Russian venture into one of the Baltic states but this is certainly not guaranteed. Only Poland, Greece (even with its poor financial condition) and Estonia spend the minimum 2% of GDP required by NATO. The UK only really meets this figure as the government has included other budgets into the equation recently.
The UK does, man for man & pound for pound, have one of the best armed forces in the world. This was the same at the start of the First World War - the British Army in 1914, due to conflicts across the Empire (especially the Boer War) was the best trained, most experienced & most mechanised. However, the Army had only limited numbers of professional soldiers & therefore was wiped out in the first few months which then resulted in 4 years of bloody trench warfare with comparatively poorly trained conscripts being thrown into the meat grinder.
The UK is in a similar state now - the Army has less than 230 Challenger 2 MBTs, the RAF has about 200 serviceable combat aircraft & and Royal Navy has only 6 Type 45 Destroyers. Even these numbers are still enough to deal with a lot of threats the UK might encounter but they're certainly not enough to deter Russia.
The Russians have recently shown off their T-14 tank which is revolutionary compared to most NATO models. The Russian economy has struggled due to Western sanctions so their aim to buy over 2000 T-14s may be unrealistic. However, Trump could ease sanctions on Russia which would certainly help the Russian economy.
Some observers have proclaimed that tanks are now obsolete due to the latest anti-tank guided missiles like Javelin & Brimstone. However, countermeasures against these missiles have also improved with both soft & hard kill protection systems that UK tanks don't have but the T-14s do. It's also fair to assume that NATO wouldn't be able to obtain air superiority in a situation that saw Russian forces strike out at the Baltic nations in time to actually stop an invasion. I can understand that large numbers of tanks aren't exactly essential for an island nation but due to our NATO commitments then I'd argue more are needed. Japan, for example, has almost 700 tanks.
It's be very embarrassing that the UK totally lost its airborne anti-submarine warfare capability after the SDSR in 2010. The 2015 SDSR has acknowledged this error & the RAF will now get American made P-8 maritime patrol aircraft. However, these are still several years from away from entering service.