NHS GPs are extremely unlikely to prescribe Wegovy or Mounjaro for weight loss alone. You would normally have to be very morbidly obese (>35 BMI), have a weight related health condition and have gone through the first tiers of NHS weight loss methods and not been successful.
So the people using these injections are nearly always going to be getting a private prescription. That could be from a private GP if you have private medical insurance, but most commonly it is via online pharmacies. You complete an online consultation form, provide photos to show your physique and also provide ID. The request is then checked by a doctor and approved. You are then sent the injections.
To be approved, you would have a BMI of over 30, or over 27 and a weight related health condition. Certain ethnic minorities might be approved at a BMI of >27 too.
If you have diabetes, your diabetes HCP might consider prescribing, in various circumstances so it could be worth discussing with them.
Side effects, well, there are mild ones that are ok to manage, and rarely some more serious ones. Mild side effects are mostly gastric and can be managed by adjusting what you eat and/or reducing the dose you're taking. If you are very obese then the risk of mild side effects is outweighed by the need to lose weight. All prescription medication will have possible side effects, this is no different to any other medication.