El C,
It is possible to become military reservist whilst employed as an airline pilot. However, the feasibility of doing both very much depends on how your roster fits in with your intended reservist commitment. The attitude of your employer and their associated HR policy towards reservists is also very significant.
For example if you were working for a charter operator and are interested in joining a reserve unit that train solely at weekends, then it would be unlikely that you will be able to make the arrangement work. However, if you were flying for a lo-co with, say, a 6/5 roster and your unit had chance to use you during those 5 days then it could be an entirely workable proposition. Clearly there are many more variations of airline job and reserve engagement than I have outlined; you have to decide what could work for you, your employer and your unit.
Firstly you need to identify which of the reserve force units that you are able to join. You can find websites for the TA, RAuxAF and RNR that will tell you all about the roles, locations and joining process. Once you have found and made contact with a unit that suits , I would suggest that the next step would be to get hold of a copy of a few months of typical line roster from your prospective airline employer. This will enable you and the unit to decide if your normal circumstances would allow a realistic contribution. It is also worth finding out from your current/future employer if they have any HR policies regarding military reservists. A supportive employer might grant some extra leave to allow you a period of continuous training.
It is worth noting that your current employer will be informed by MoD that you have joined the reserves. Moreover, some reservist roles with a high deployment readiness actually require the employer to give their continued consent on an annual basis. In general, your employer does not have to allow you to become a reservist, but cannot dismiss you on the grounds of your reservist commitment. This is a good website to find out more about your rights and responsibilities.
SaBRE - Support for Britain's Reservists and Employers - TA, RAuxAF, RNR and RMR
Above all, remember that you are taking on a commitment that could involve mobilization for operations. How would that affect your fledging career as an airline pilot?
Good luck.