The bloating and constipation may not be gluten but from a sugar (fructans) that's found in wheat. It's also in loads of other foods.
www.monashfodmap.com/blog/avoiding-wheat-how-strict-on-low-fodmap_10/
www.monashfodmap.com/blog/fructans-fodmap-reintroduction/
It could also be something completely different causing the bloatedness.
How soon do you get bloated after eating?
It can take around 12-24 hrs for food to reach the gut so food you are currently eating when the bloating occurs may not be the food you are actually responding to. Eating fires up the digestive system so your gut starts processing what is in there and you get a reaction if it's something you are intolerant to.
As others have said if you think you might be coeliac, you need to be eating a good amount of gluten for 6 weeks before you can be tested. So it's worth asking for the test before you give up gluten as it can be difficult to do once you know it's making you unwell. If you are coeliac you will need to follow a very strict item free diet avoiding cross contamination for the rest of your life. Not something to undertake unless you are certain.
I'm just going to caveat that by saying that a coeliac blood test is not 100% accurate.
celiac.org/about-celiac-disease/screening-and-diagnosis/screening/
A bowel biopsy is more definitive but you are unlikely to get that on the NHS straight out of the gate. They may want you to try other things first.
Have you spoken to your doctor? There are a lot of conditions that can cause constipation and bloating.
If you have and they've ruled everything else out and they think IBS (IBS is a diagnosis of exclusion and should not be diagnosed without everything else being ruled out first), take a look at low FODMAP.
I cured my IBS with it (ie it enabled me to identify exactly what I was intolerant to and have no symptoms now - thankfully for me there are enzymes I can take to help me process the foods I was intolerant to so I basically eat almost anything now).
It's full on. So do a lot of research before you start and get prepared.
A good starting place is here:
www.monashfodmap.com
These apps are also helpful:
www.monashfodmap.com/ibs-central/i-have-ibs/get-the-app/
www.kcl.ac.uk/research/work-with-kings/ip-licensing/fodmap-app
It's complicated so it's recommended you do it under a dietician. But you do need a dietician who knows what they are talking about! I did it myself but I am happy to do extensive research and am a bit if a stickler for detail so if you are not that way inclined you might want to see if you can get a dietician referral.
You follow the diet for 6-8 weeks until symptoms subside and then follow a specific reintroduction programme where you slowly reintroduce the sugar groups and monitor symptoms.
The diet is very restrictive so you should not stay on it for too long or it will start to negatively affect your gut microbiome. If symptoms don't subside after 6-8 weeks then it's probably not fodmaps if you are sure you've followed it correctly.
Good luck.