What exactly is the setup you are on? Trainer? Static bike?
A different max is expected but it should be closer than you are getting.
You did get to 153 or something like it on the last few efforts. HR was still climbing when you stopped, what made you stop? Your spinning a fast cadence at 110 odd the indicated speed is low for what you would expect with real world type resistance etc. e.g. about 20mph where-as a real world flat out minute I would hope you could be quicker than that.
I found it just takes more cycling to be able to increase the HR on the bike. Doing more hard sessions gradually increased how high my HR could go. It's still lower than what I'd expect for the same effort running but I can easily get above the 110bpm I used to get
Road bike on turbo trainer - the last few efforts today were 50 seconds at a tempo effort followed by 10 seconds all out. That's probably the highest I've gotten it on the turbo, although I've managed similar on a spin bike in the gym.. I find that a faster cadence feels much more sustainable, lower cadence gets my legs burning faster. It could well just be a practise thing.
What are the recoveries like? When i run i find float recoveries hellish but theyre the only way i can get a good session alternative training. Wasnt sure about not having much between recovery and interval pace at first but it makes the session cumulative and at least by the end im working at ~95% HR (granted im usually on the eliptical but im guessing similar would apply for a bike?)
Overheating is a big limiting factor on a turbo trainer - reducing the ambient temperature helps a bit (e.g. if you can put the turbo in a garage) but better if you can get a (big) fan to actually blow air over you.
Like you (but at a lower level, obv) I find it difficult to get my HR up on a bike or TT but the best option for me is relatively short intervals with even shorter recoveries: 40 seconds on, 20 off (the last five seconds of which are winding up for the next interval). I find HR builds through the session - from 135-140 on the first couple up to 155-160 after 15-20 reps - and also that my (burning) legs recover very quickly (so you can always do more reps than you think) whilst HR doesn't drop too much in the short recovery.
To take it to extremes, a tabata session (8*20s/10s) can be quite effective on the turbo, but you need to get v well warmed up, push a massive gear/huge resistance
and be prepared to smash it - if you aren't beginning to black out by the 7th rep you aren't doing it right :-)
I also reckon from having done it the last few months that GPS watches/HRM's just don't seem to track HR as accurately when static or x-training, my watch can say 100 but if I check my pulse it's going like crazy so I think there's definitely an element of mechanical error there too. If you're working hard by perceived effort then you're still getting the benefits regardless of the stats