You will likely be found at fault. Bicycle lanes are considered a lane of traffic that supports only one type of traffic for extended periods of time (i.e., bicycles). As such, by your description you signaled your intent, then turned right across another lane of traffic and collided with another vehicle already occupying that lane. It is your job to ensure a lane of traffic is clear before entering it.
If you were talking about two car lanes I think there would little if any questions regarding fault, but because we are talking about cycling many motorists assume it is the job of the cyclist to get out of the way because they are more vulnerable. While true from a practical perspective, this perspective is not supported legally.
Further confusing matters, in some areas (e.g., North America) drivers are required to merge over into the bike lane (when safe) before turning, while other areas of the world do not allow any merging and instead give the cyclists the full priority at all times. Regardless, both approaches would find legal fault with the motorist turning across another vehicle lane (i.e., the cycling lane) when it was not clear to do so.
So what should you have done (North America)?
Generally in most North American jurisdictions if the bicycle lane is clear you should first merge over into that lane before making a right turn (see diagram below). This merge area is often indicated with dashed lines separating the car and bicycle lanes. If you safely merge over, then you have physical position over the cyclist and they should wait until you complete your turn before passing if they wish to remain in the bicycle lane, otherwise they will need to change lane into the main traffic lane to the left to pass (assuming of course a jurisdiction that drives on the right).
While you had your signal on you were still in the lane to the left of the cycling lane (i.e., left of another vehicle lane) and as such you had only signaled your intent. Until you safely enter the cycling lane, to complete your right turn, it is expectant upon you to check for traffic before moving to the right.
From San Fransisco Bicycle Coalition
Other North American Jurisdictions supporting this approach
I chose the figure above because it clearly communicated the concept. Some might be concerned that this only applies to San Francisco, but similar rules apply in many other North American regions. Some further examples are given below:
Ontario, Canada
Diagram 2-12
Bike lanes are reserved for cyclists. They are typically marked by a solid white line. Sometimes you will need to enter or cross a bike lane to turn right at a corner or driveway. (See Diagram 2-12) Take extra care when you do this. Enter the bike lane only after ensuring that you can do so safely, and then make the turn.
-The Official Ministry of Transportation (MTO) Driver's Handbook
British Columbia, Canada
Where there is a broken line marking the cycle lane at the approach to the intersection the driver may move over to the curb into the cycle lane prior to making the right turn in the way that most people are used to. Drivers doing so are making a lane change and must yield to cycle traffic in the bicycle lane before moving over! In this situation, the cyclist must wait behind the vehicle until after the turn is made to clear the cycle lane.
-DriveSmart BC
California, USA
21717.Whenever it is necessary for the driver of a motor vehicle to cross a bicycle lane that is adjacent to his lane of travel to make a turn, the driver shall drive the motor vehicle into the bicycle lane prior to making the turn and shall make the turn pursuant to Section 22100.
California Law
European Jurisdictions
European jurisdictions give bicycle lanes legal vehicle status, but what seems less clear is if there is support for motor vehicles merging across bicycle lanes before a right turn (if driving on the right). From the comments it appears most European nations give cyclists the absolute priority (i.e., the driver needs to wait until there is no cycling lane traffic before turning). Explicit examples are still needed.
Germany (by sleske)
In Germany, you would be at fault (probably fully). If you make a turn (no matter whether right or left), the traffic behind you in other lanes has priority, and you must make sure you do not obstruct or endanger it before turning.
This is based on §9 Straßenverkehrsordnung (German traffic code):
Abbiegen, Wenden und Rückwärtsfahren [...]
Vor dem Einordnen und nochmals vor dem Abbiegen ist auf den
nachfolgenden Verkehr zu achten; vor dem Abbiegen ist es dann nicht
nötig, wenn eine Gefährdung nachfolgenden Verkehrs ausgeschlossen ist.
Translation (mine, no guarantees):
Turning and reversing [...] Before changing lanes and again before turning,
drivers must look out for following traffic; this is not necessary before turning
if a danger to following traffic is not possible.
Southwestern Pacific
New Zealand (left hand traffic)
In New Zealand cyclists appear to have absolute priority as turning motorists are not allowed to merge across the bike lane to make a turn and are required to wait until the bike lane is clear:
When there’s a cycle lane at a set of lights, you must not wait in it for the light to turn green, and you must not drive in it as you approach the lights (unless you’re riding a bicycle). Motorbikes must not use this lane, either, but they can filter between stationary traffic.
Cycles have the right-of-way in this lane. If there are cycles waiting to go forward, you must give way to them before you can turn left.
The car is waiting in the wrong position, blocking the cycleway
This shows the correct position with the left-hand wheels of the car not encroaching into the cycle lane
-Driving Tests NZ
