Garmin 310XT with HRM is about £122 at Amazon at the moment – there’s a newer high-end triathlon model so it’s discounted now. It’s a little bulky but will do all the tracking stuff absolutely fine, and is waterproof enough to swim with. The watches work OK on the bike – either Garmin will sell you a “mount” or I used to use a cut-down old grip to bulk out the handlebar to wrap it around.
Else you can get a Forerunner 15 or 210 which has a much slimmer profile but shows far less info at once and less functionality – eg. the 310XT can be used to follow a course.
Or if the goal is more bike-focussed then an Edge (probably 500 is in budget). You can get a wrist strap with the twist mount for them, but it’s a bit bulky – or just chuck it in a pocket when you’re running.
tldr: buy a Garmin Forerunner 310XT.